Calculus Examples

$- \frac{1}{3} x^{6} - 3 x^{5} - \frac{15}{2} x^{4}$

Step 1

Write $- \frac{1}{3} x^{6} - 3 x^{5} - \frac{15}{2} x^{4}$ as a function.

$f (x) = - \frac{1}{3} x^{6} - 3 x^{5} - \frac{15}{2} x^{4}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $- \frac{1}{3} x^{6} - 3 x^{5} - \frac{15}{2} x^{4}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{3} x^{6}] + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$ .

$\frac{d}{d x} [- \frac{1}{3} x^{6}] + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [- \frac{1}{3} x^{6}]$ .

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Step 2.1.2.1

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{1}{3} x^{6}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x^{6}]$ .

$- \frac{1}{3} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$- \frac{1}{3} (6 x^{5}) + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2.3

Multiply $6$ by $- 1$ .

$- 6 (\frac{1}{3}) x^{5} + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2.4

Combine $- 6$ and $\frac{1}{3}$ .

$\frac{- 6}{3} x^{5} + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2.5

Combine $\frac{- 6}{3}$ and $x^{5}$ .

$\frac{- 6 x^{5}}{3} + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2.6

Cancel the common factor of $- 6$ and $3$ .

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Step 2.1.2.6.1

Factor $3$ out of $- 6 x^{5}$ .

$\frac{3 (- 2 x^{5})}{3} + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2.6.2

Cancel the common factors.

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Step 2.1.2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (- 2 x^{5})}{3 (1)} + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2.6.2.2

Cancel the common factor.

$\frac{3 (- 2 x^{5})}{3 \cdot 1} + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2.6.2.3

Rewrite the expression.

$\frac{- 2 x^{5}}{1} + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.2.6.2.4

Divide $- 2 x^{5}$ by $1$ .

$- 2 x^{5} + \frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 3 x^{5}]$ .

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Step 2.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{5}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{5}]$ .

$- 2 x^{5} - 3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- 2 x^{5} - 3 (5 x^{4}) + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.3.3

Multiply $5$ by $- 3$ .

$- 2 x^{5} - 15 x^{4} + \frac{d}{d x} [- \frac{15}{2} x^{4}]$

Step 2.1.4

Evaluate $\frac{d}{d x} [- \frac{15}{2} x^{4}]$ .

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Step 2.1.4.1

Since $- \frac{15}{2}$ is constant with respect to $x$ , the derivative of $- \frac{15}{2} x^{4}$ with respect to $x$ is $- \frac{15}{2} \frac{d}{d x} [x^{4}]$ .

$- 2 x^{5} - 15 x^{4} - \frac{15}{2} \frac{d}{d x} [x^{4}]$

Step 2.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 2 x^{5} - 15 x^{4} - \frac{15}{2} (4 x^{3})$

Step 2.1.4.3

Multiply $4$ by $- 1$ .

$- 2 x^{5} - 15 x^{4} - 4 (\frac{15}{2}) x^{3}$

Step 2.1.4.4

Combine $- 4$ and $\frac{15}{2}$ .

$- 2 x^{5} - 15 x^{4} + \frac{- 4 \cdot 15}{2} x^{3}$

Step 2.1.4.5

Multiply $- 4$ by $15$ .

$- 2 x^{5} - 15 x^{4} + \frac{- 60}{2} x^{3}$

Step 2.1.4.6

Combine $\frac{- 60}{2}$ and $x^{3}$ .

$- 2 x^{5} - 15 x^{4} + \frac{- 60 x^{3}}{2}$

Step 2.1.4.7

Cancel the common factor of $- 60$ and $2$ .

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Step 2.1.4.7.1

Factor $2$ out of $- 60 x^{3}$ .

$- 2 x^{5} - 15 x^{4} + \frac{2 (- 30 x^{3})}{2}$

Step 2.1.4.7.2

Cancel the common factors.

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Step 2.1.4.7.2.1

Factor $2$ out of $2$ .

$- 2 x^{5} - 15 x^{4} + \frac{2 (- 30 x^{3})}{2 (1)}$

Step 2.1.4.7.2.2

Cancel the common factor.

$- 2 x^{5} - 15 x^{4} + \frac{2 (- 30 x^{3})}{2 \cdot 1}$

Step 2.1.4.7.2.3

Rewrite the expression.

$- 2 x^{5} - 15 x^{4} + \frac{- 30 x^{3}}{1}$

Step 2.1.4.7.2.4

Divide $- 30 x^{3}$ by $1$ .

$f' (x) = - 2 x^{5} - 15 x^{4} - 30 x^{3}$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $- 2 x^{5} - 15 x^{4} - 30 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{5}] + \frac{d}{d x} [- 15 x^{4}] + \frac{d}{d x} [- 30 x^{3}]$ .

$\frac{d}{d x} [- 2 x^{5}] + \frac{d}{d x} [- 15 x^{4}] + \frac{d}{d x} [- 30 x^{3}]$

Step 2.2.2

Evaluate $\frac{d}{d x} [- 2 x^{5}]$ .

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Step 2.2.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{5}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{5}]$ .

$- 2 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 15 x^{4}] + \frac{d}{d x} [- 30 x^{3}]$

Step 2.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- 2 (5 x^{4}) + \frac{d}{d x} [- 15 x^{4}] + \frac{d}{d x} [- 30 x^{3}]$

Step 2.2.2.3

Multiply $5$ by $- 2$ .

$- 10 x^{4} + \frac{d}{d x} [- 15 x^{4}] + \frac{d}{d x} [- 30 x^{3}]$

Step 2.2.3

Evaluate $\frac{d}{d x} [- 15 x^{4}]$ .

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Step 2.2.3.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15 x^{4}$ with respect to $x$ is $- 15 \frac{d}{d x} [x^{4}]$ .

$- 10 x^{4} - 15 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 30 x^{3}]$

Step 2.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 10 x^{4} - 15 (4 x^{3}) + \frac{d}{d x} [- 30 x^{3}]$

Step 2.2.3.3

Multiply $4$ by $- 15$ .

$- 10 x^{4} - 60 x^{3} + \frac{d}{d x} [- 30 x^{3}]$

Step 2.2.4

Evaluate $\frac{d}{d x} [- 30 x^{3}]$ .

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Step 2.2.4.1

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30 x^{3}$ with respect to $x$ is $- 30 \frac{d}{d x} [x^{3}]$ .

$- 10 x^{4} - 60 x^{3} - 30 \frac{d}{d x} [x^{3}]$

Step 2.2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- 10 x^{4} - 60 x^{3} - 30 (3 x^{2})$

Step 2.2.4.3

Multiply $3$ by $- 30$ .

$f'' (x) = - 10 x^{4} - 60 x^{3} - 90 x^{2}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- 10 x^{4} - 60 x^{3} - 90 x^{2}$ .

$- 10 x^{4} - 60 x^{3} - 90 x^{2}$

Step 3

Set the second derivative equal to $0$ then solve the equation $- 10 x^{4} - 60 x^{3} - 90 x^{2} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- 10 x^{4} - 60 x^{3} - 90 x^{2} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- 10 x^{2}$ out of $- 10 x^{4} - 60 x^{3} - 90 x^{2}$ .

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Step 3.2.1.1

Factor $- 10 x^{2}$ out of $- 10 x^{4}$ .

$- 10 x^{2} x^{2} - 60 x^{3} - 90 x^{2} = 0$

Step 3.2.1.2

Factor $- 10 x^{2}$ out of $- 60 x^{3}$ .

$- 10 x^{2} x^{2} - 10 x^{2} (6 x) - 90 x^{2} = 0$

Step 3.2.1.3

Factor $- 10 x^{2}$ out of $- 90 x^{2}$ .

$- 10 x^{2} x^{2} - 10 x^{2} (6 x) - 10 x^{2} \cdot 9 = 0$

Step 3.2.1.4

Factor $- 10 x^{2}$ out of $- 10 x^{2} (x^{2}) - 10 x^{2} (6 x)$ .

$- 10 x^{2} (x^{2} + 6 x) - 10 x^{2} \cdot 9 = 0$

Step 3.2.1.5

Factor $- 10 x^{2}$ out of $- 10 x^{2} (x^{2} + 6 x) - 10 x^{2} (9)$ .

$- 10 x^{2} (x^{2} + 6 x + 9) = 0$

Step 3.2.2

Factor using the perfect square rule.

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Step 3.2.2.1

Rewrite $9$ as $3^{2}$ .

$- 10 x^{2} (x^{2} + 6 x + 3^{2}) = 0$

Step 3.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 3.2.2.3

Rewrite the polynomial.

$- 10 x^{2} (x^{2} + 2 \cdot x \cdot 3 + 3^{2}) = 0$

Step 3.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

$- 10 x^{2} {(x + 3)}^{2} = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

${(x + 3)}^{2} = 0$

Step 3.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 3.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 3.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 3.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.5

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set ${(x + 3)}^{2}$ equal to $0$ .

${(x + 3)}^{2} = 0$

Step 3.5.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

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Step 3.5.2.1

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.5.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.6

The final solution is all the values that make $- 10 x^{2} {(x + 3)}^{2} = 0$ true.

$x = 0, - 3$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $0$ in $f (x) = - \frac{1}{3} x^{6} - 3 x^{5} - \frac{15}{2} x^{4}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \frac{1}{3} \cdot {(0)}^{6} - 3 {(0)}^{5} - \frac{15}{2} \cdot {(0)}^{4}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = - \frac{1}{3} \cdot 0 - 3 {(0)}^{5} - \frac{15}{2} \cdot {(0)}^{4}$

Step 4.1.2.1.2

Multiply $- \frac{1}{3} \cdot 0$ .

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Step 4.1.2.1.2.1

Multiply $0$ by $- 1$ .

$f (0) = 0 (\frac{1}{3}) - 3 {(0)}^{5} - \frac{15}{2} \cdot {(0)}^{4}$

Step 4.1.2.1.2.2

Multiply $0$ by $\frac{1}{3}$ .

$f (0) = 0 - 3 {(0)}^{5} - \frac{15}{2} \cdot {(0)}^{4}$

Step 4.1.2.1.3

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - 3 \cdot 0 - \frac{15}{2} \cdot {(0)}^{4}$

Step 4.1.2.1.4

Multiply $- 3$ by $0$ .

$f (0) = 0 + 0 - \frac{15}{2} \cdot {(0)}^{4}$

Step 4.1.2.1.5

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 0 - \frac{15}{2} \cdot 0$

Step 4.1.2.1.6

Multiply $- \frac{15}{2} \cdot 0$ .

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Step 4.1.2.1.6.1

Multiply $0$ by $- 1$ .

$f (0) = 0 + 0 + 0 (\frac{15}{2})$

Step 4.1.2.1.6.2

Multiply $0$ by $\frac{15}{2}$ .

$f (0) = 0 + 0 + 0$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 4.1.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 4.1.2.3

The final answer is $0$ .

$0$

Step 4.2

The point found by substituting $0$ in $f (x) = - \frac{1}{3} x^{6} - 3 x^{5} - \frac{15}{2} x^{4}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4.3

Substitute $- 3$ in $f (x) = - \frac{1}{3} x^{6} - 3 x^{5} - \frac{15}{2} x^{4}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - \frac{1}{3} \cdot {(- 3)}^{6} - 3 {(- 3)}^{5} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $- 3$ to the power of $6$ .

$f (- 3) = - \frac{1}{3} \cdot 729 - 3 {(- 3)}^{5} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.2

Cancel the common factor of $3$ .

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Step 4.3.2.1.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$f (- 3) = \frac{- 1}{3} \cdot 729 - 3 {(- 3)}^{5} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.2.2

Factor $3$ out of $729$ .

$f (- 3) = \frac{- 1}{3} \cdot (3 (243)) - 3 {(- 3)}^{5} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.2.3

Cancel the common factor.

$f (- 3) = \frac{- 1}{3} \cdot (3 \cdot 243) - 3 {(- 3)}^{5} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.2.4

Rewrite the expression.

$f (- 3) = - 1 \cdot 243 - 3 {(- 3)}^{5} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.3

Multiply $- 1$ by $243$ .

$f (- 3) = - 243 - 3 {(- 3)}^{5} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.4

Multiply $- 3$ by ${(- 3)}^{5}$ by adding the exponents.

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Step 4.3.2.1.4.1

Multiply $- 3$ by ${(- 3)}^{5}$ .

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Step 4.3.2.1.4.1.1

Raise $- 3$ to the power of $1$ .

$f (- 3) = - 243 + (- 3) {(- 3)}^{5} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 3) = - 243 + {(- 3)}^{1 + 5} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.4.2

Add $1$ and $5$ .

$f (- 3) = - 243 + {(- 3)}^{6} - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.5

Raise $- 3$ to the power of $6$ .

$f (- 3) = - 243 + 729 - \frac{15}{2} \cdot {(- 3)}^{4}$

Step 4.3.2.1.6

Raise $- 3$ to the power of $4$ .

$f (- 3) = - 243 + 729 - \frac{15}{2} \cdot 81$

Step 4.3.2.1.7

Multiply $- \frac{15}{2} \cdot 81$ .

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Step 4.3.2.1.7.1

Multiply $81$ by $- 1$ .

$f (- 3) = - 243 + 729 - 81 (\frac{15}{2})$

Step 4.3.2.1.7.2

Combine $- 81$ and $\frac{15}{2}$ .

$f (- 3) = - 243 + 729 + \frac{- 81 \cdot 15}{2}$

Step 4.3.2.1.7.3

Multiply $- 81$ by $15$ .

$f (- 3) = - 243 + 729 + \frac{- 1215}{2}$

Step 4.3.2.1.8

Move the negative in front of the fraction.

$f (- 3) = - 243 + 729 - \frac{1215}{2}$

Step 4.3.2.2

Find the common denominator.

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Step 4.3.2.2.1

Write $- 243$ as a fraction with denominator $1$ .

$f (- 3) = \frac{- 243}{1} + 729 - \frac{1215}{2}$

Step 4.3.2.2.2

Multiply $\frac{- 243}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{- 243}{1} \cdot \frac{2}{2} + 729 - \frac{1215}{2}$

Step 4.3.2.2.3

Multiply $\frac{- 243}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{- 243 \cdot 2}{2} + 729 - \frac{1215}{2}$

Step 4.3.2.2.4

Write $729$ as a fraction with denominator $1$ .

$f (- 3) = \frac{- 243 \cdot 2}{2} + \frac{729}{1} - \frac{1215}{2}$

Step 4.3.2.2.5

Multiply $\frac{729}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{- 243 \cdot 2}{2} + \frac{729}{1} \cdot \frac{2}{2} - \frac{1215}{2}$

Step 4.3.2.2.6

Multiply $\frac{729}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{- 243 \cdot 2}{2} + \frac{729 \cdot 2}{2} - \frac{1215}{2}$

Step 4.3.2.3

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 243 \cdot 2 + 729 \cdot 2 - 1215}{2}$

Step 4.3.2.4

Simplify each term.

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Step 4.3.2.4.1

Multiply $- 243$ by $2$ .

$f (- 3) = \frac{- 486 + 729 \cdot 2 - 1215}{2}$

Step 4.3.2.4.2

Multiply $729$ by $2$ .

$f (- 3) = \frac{- 486 + 1458 - 1215}{2}$

Step 4.3.2.5

Simplify the expression.

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Step 4.3.2.5.1

Add $- 486$ and $1458$ .

$f (- 3) = \frac{972 - 1215}{2}$

Step 4.3.2.5.2

Subtract $1215$ from $972$ .

$f (- 3) = \frac{- 243}{2}$

Step 4.3.2.5.3

Move the negative in front of the fraction.

$f (- 3) = - \frac{243}{2}$

Step 4.3.2.6

The final answer is $- \frac{243}{2}$ .

$- \frac{243}{2}$

Step 4.4

The point found by substituting $- 3$ in $f (x) = - \frac{1}{3} x^{6} - 3 x^{5} - \frac{15}{2} x^{4}$ is $(- 3, - \frac{243}{2})$ . This point can be an inflection point.

$(- 3, - \frac{243}{2})$

Step 4.5

Determine the points that could be inflection points.

$(0, 0), (- 3, - \frac{243}{2})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 3) \cup (- 3, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 3)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 3.1$ in the expression.

$f'' (- 3.1) = - 10 {(- 3.1)}^{4} - 60 {(- 3.1)}^{3} - 90 {(- 3.1)}^{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 3.1$ to the power of $4$ .

$f'' (- 3.1) = - 10 \cdot 92.3521 - 60 {(- 3.1)}^{3} - 90 {(- 3.1)}^{2}$

Step 6.2.1.2

Multiply $- 10$ by $92.3521$ .

$f'' (- 3.1) = - 923.521 - 60 {(- 3.1)}^{3} - 90 {(- 3.1)}^{2}$

Step 6.2.1.3

Raise $- 3.1$ to the power of $3$ .

$f'' (- 3.1) = - 923.521 - 60 \cdot - 29.791 - 90 {(- 3.1)}^{2}$

Step 6.2.1.4

Multiply $- 60$ by $- 29.791$ .

$f'' (- 3.1) = - 923.521 + 1787.46 - 90 {(- 3.1)}^{2}$

Step 6.2.1.5

Raise $- 3.1$ to the power of $2$ .

$f'' (- 3.1) = - 923.521 + 1787.46 - 90 \cdot 9.61$

Step 6.2.1.6

Multiply $- 90$ by $9.61$ .

$f'' (- 3.1) = - 923.521 + 1787.46 - 864.9$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 923.521$ and $1787.46$ .

$f'' (- 3.1) = 863.939 - 864.9$

Step 6.2.2.2

Subtract $864.9$ from $863.939$ .

$f'' (- 3.1) = - 0.961$

Step 6.2.3

The final answer is $- 0.961$ .

$- 0.961$

Step 6.3

At $- 3.1$ , the second derivative is $- 0.961$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 3)$

Decreasing on $(- \infty, - 3)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 3, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

$f'' (- \frac{3}{2}) = - 10 {(- \frac{3}{2})}^{4} - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$f'' (- \frac{3}{2}) = - 10 ({(- 1)}^{4} {(\frac{3}{2})}^{4}) - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$f'' (- \frac{3}{2}) = - 10 ({(- 1)}^{4} (\frac{3^{4}}{2^{4}})) - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.2

Raise $- 1$ to the power of $4$ .

$f'' (- \frac{3}{2}) = - 10 (1 (\frac{3^{4}}{2^{4}})) - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.3

Multiply $\frac{3^{4}}{2^{4}}$ by $1$ .

$f'' (- \frac{3}{2}) = - 10 \frac{3^{4}}{2^{4}} - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.4

Raise $3$ to the power of $4$ .

$f'' (- \frac{3}{2}) = - 10 \frac{81}{2^{4}} - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.5

Raise $2$ to the power of $4$ .

$f'' (- \frac{3}{2}) = - 10 (\frac{81}{16}) - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.6

Cancel the common factor of $2$ .

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Step 7.2.1.6.1

Factor $2$ out of $- 10$ .

$f'' (- \frac{3}{2}) = 2 (- 5) (\frac{81}{16}) - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.6.2

Factor $2$ out of $16$ .

$f'' (- \frac{3}{2}) = 2 \cdot (- 5 \frac{81}{2 \cdot 8}) - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.6.3

Cancel the common factor.

$f'' (- \frac{3}{2}) = 2 \cdot (- 5 \frac{81}{2 \cdot 8}) - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.6.4

Rewrite the expression.

$f'' (- \frac{3}{2}) = - 5 (\frac{81}{8}) - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.7

Combine $- 5$ and $\frac{81}{8}$ .

$f'' (- \frac{3}{2}) = \frac{- 5 \cdot 81}{8} - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.8

Multiply $- 5$ by $81$ .

$f'' (- \frac{3}{2}) = \frac{- 405}{8} - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.9

Move the negative in front of the fraction.

$f'' (- \frac{3}{2}) = - \frac{405}{8} - 60 {(- \frac{3}{2})}^{3} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.2.1.10.1

Apply the product rule to $- \frac{3}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} - 60 ({(- 1)}^{3} {(\frac{3}{2})}^{3}) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.10.2

Apply the product rule to $\frac{3}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} - 60 ({(- 1)}^{3} (\frac{3^{3}}{2^{3}})) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.11

Raise $- 1$ to the power of $3$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} - 60 (- \frac{3^{3}}{2^{3}}) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.12

Raise $3$ to the power of $3$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} - 60 (- \frac{27}{2^{3}}) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.13

Raise $2$ to the power of $3$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} - 60 (- \frac{27}{8}) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.14

Cancel the common factor of $4$ .

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Step 7.2.1.14.1

Move the leading negative in $- \frac{27}{8}$ into the numerator.

$f'' (- \frac{3}{2}) = - \frac{405}{8} - 60 (\frac{- 27}{8}) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.14.2

Factor $4$ out of $- 60$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + 4 (- 15) (\frac{- 27}{8}) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.14.3

Factor $4$ out of $8$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + 4 \cdot (- 15 \frac{- 27}{4 \cdot 2}) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.14.4

Cancel the common factor.

$f'' (- \frac{3}{2}) = - \frac{405}{8} + 4 \cdot (- 15 \frac{- 27}{4 \cdot 2}) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.14.5

Rewrite the expression.

$f'' (- \frac{3}{2}) = - \frac{405}{8} - 15 (\frac{- 27}{2}) - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.15

Combine $- 15$ and $\frac{- 27}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{- 15 \cdot - 27}{2} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.16

Multiply $- 15$ by $- 27$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} - 90 {(- \frac{3}{2})}^{2}$

Step 7.2.1.17

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.2.1.17.1

Apply the product rule to $- \frac{3}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} - 90 ({(- 1)}^{2} {(\frac{3}{2})}^{2})$

Step 7.2.1.17.2

Apply the product rule to $\frac{3}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} - 90 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}}))$

Step 7.2.1.18

Raise $- 1$ to the power of $2$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} - 90 (1 (\frac{3^{2}}{2^{2}}))$

Step 7.2.1.19

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} - 90 \frac{3^{2}}{2^{2}}$

Step 7.2.1.20

Raise $3$ to the power of $2$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} - 90 \frac{9}{2^{2}}$

Step 7.2.1.21

Raise $2$ to the power of $2$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} - 90 (\frac{9}{4})$

Step 7.2.1.22

Cancel the common factor of $2$ .

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Step 7.2.1.22.1

Factor $2$ out of $- 90$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} + 2 (- 45) (\frac{9}{4})$

Step 7.2.1.22.2

Factor $2$ out of $4$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} + 2 \cdot (- 45 \frac{9}{2 \cdot 2})$

Step 7.2.1.22.3

Cancel the common factor.

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} + 2 \cdot (- 45 \frac{9}{2 \cdot 2})$

Step 7.2.1.22.4

Rewrite the expression.

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} - 45 (\frac{9}{2})$

Step 7.2.1.23

Combine $- 45$ and $\frac{9}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} + \frac{- 45 \cdot 9}{2}$

Step 7.2.1.24

Multiply $- 45$ by $9$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} + \frac{- 405}{2}$

Step 7.2.1.25

Move the negative in front of the fraction.

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{405}{2} - \frac{405}{2}$

Step 7.2.2

Combine fractions.

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Step 7.2.2.1

Combine the numerators over the common denominator.

$f'' (- \frac{3}{2}) = \frac{- 405}{8} + \frac{405 - 405}{2}$

Step 7.2.2.2

Subtract $405$ from $405$ .

$f'' (- \frac{3}{2}) = \frac{- 405}{8} + \frac{0}{2}$

Step 7.2.3

Simplify each term.

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Step 7.2.3.1

Move the negative in front of the fraction.

$f'' (- \frac{3}{2}) = - \frac{405}{8} + \frac{0}{2}$

Step 7.2.3.2

Divide $0$ by $2$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8} + 0$

Step 7.2.4

Add $- \frac{405}{8}$ and $0$ .

$f'' (- \frac{3}{2}) = - \frac{405}{8}$

Step 7.2.5

The final answer is $- \frac{405}{8}$ .

$- \frac{405}{8}$

Step 7.3

At $- \frac{3}{2}$ , the second derivative is $- 50.625$ . Since this is negative, the second derivative is decreasing on the interval $(- 3, 0)$

Decreasing on $(- 3, 0)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = - 10 {(0.1)}^{4} - 60 {(0.1)}^{3} - 90 {(0.1)}^{2}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $0.1$ to the power of $4$ .

$f'' (0.1) = - 10 \cdot 0.0001 - 60 {(0.1)}^{3} - 90 {(0.1)}^{2}$

Step 8.2.1.2

Multiply $- 10$ by $0.0001$ .

$f'' (0.1) = - 0.001 - 60 {(0.1)}^{3} - 90 {(0.1)}^{2}$

Step 8.2.1.3

Raise $0.1$ to the power of $3$ .

$f'' (0.1) = - 0.001 - 60 \cdot 0.001 - 90 {(0.1)}^{2}$

Step 8.2.1.4

Multiply $- 60$ by $0.001$ .

$f'' (0.1) = - 0.001 - 0.06 - 90 {(0.1)}^{2}$

Step 8.2.1.5

Raise $0.1$ to the power of $2$ .

$f'' (0.1) = - 0.001 - 0.06 - 90 \cdot 0.01$

Step 8.2.1.6

Multiply $- 90$ by $0.01$ .

$f'' (0.1) = - 0.001 - 0.06 - 0.9$

Step 8.2.2

Simplify by subtracting numbers.

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Step 8.2.2.1

Subtract $0.06$ from $- 0.001$ .

$f'' (0.1) = - 0.061 - 0.9$

Step 8.2.2.2

Subtract $0.9$ from $- 0.061$ .

$f'' (0.1) = - 0.961$

Step 8.2.3

The final answer is $- 0.961$ .

$- 0.961$

Step 8.3

At $0.1$ , the second derivative is $- 0.961$ . Since this is negative, the second derivative is decreasing on the interval $(0, \infty)$

Decreasing on $(0, \infty)$ since $f'' (x) < 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. There are no points on the graph that satisfy these requirements.

No Inflection Points