Calculus Examples

$- \frac{1}{15} x^{6} + 6 x^{4}$

Step 1

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

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

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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

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

Step 2.1.1.2

Evaluate $\frac{d}{d x} [- \frac{1}{15} x^{6}]$ .

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Step 2.1.1.2.1

Since $- \frac{1}{15}$ is constant with respect to $x$ , the derivative of $- \frac{1}{15} x^{6}$ with respect to $x$ is $- \frac{1}{15} \frac{d}{d x} [x^{6}]$ .

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

Multiply $6$ by $- 1$ .

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

Step 2.1.1.2.4

Combine $- 6$ and $\frac{1}{15}$ .

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

Step 2.1.1.2.5

Combine $\frac{- 6}{15}$ and $x^{5}$ .

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

Step 2.1.1.2.6

Cancel the common factor of $- 6$ and $15$ .

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Step 2.1.1.2.6.1

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

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

Step 2.1.1.2.6.2

Cancel the common factors.

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Step 2.1.1.2.6.2.1

Factor $3$ out of $15$ .

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

Step 2.1.1.2.6.2.2

Cancel the common factor.

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

Step 2.1.1.2.6.2.3

Rewrite the expression.

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

Step 2.1.1.2.7

Move the negative in front of the fraction.

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

Step 2.1.1.3

Evaluate $\frac{d}{d x} [6 x^{4}]$ .

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Step 2.1.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4}$ with respect to $x$ is $6 \frac{d}{d x} [x^{4}]$ .

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

Step 2.1.1.3.2

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

$- \frac{2 x^{5}}{5} + 6 (4 x^{3})$

Step 2.1.1.3.3

Multiply $4$ by $6$ .

$f' (x) = - \frac{2 x^{5}}{5} + 24 x^{3}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $- \frac{2 x^{5}}{5} + 24 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- \frac{2 x^{5}}{5}] + \frac{d}{d x} [24 x^{3}]$ .

$\frac{d}{d x} [- \frac{2 x^{5}}{5}] + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$- \frac{2}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.2

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

$- \frac{2}{5} (5 x^{4}) + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.3

Multiply $5$ by $- 1$ .

$- 5 (\frac{2}{5}) x^{4} + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.4

Combine $- 5$ and $\frac{2}{5}$ .

$\frac{- 5 \cdot 2}{5} x^{4} + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.5

Multiply $- 5$ by $2$ .

$\frac{- 10}{5} x^{4} + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.6

Combine $\frac{- 10}{5}$ and $x^{4}$ .

$\frac{- 10 x^{4}}{5} + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.7

Cancel the common factor of $- 10$ and $5$ .

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Step 2.1.2.2.7.1

Factor $5$ out of $- 10 x^{4}$ .

$\frac{5 (- 2 x^{4})}{5} + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.7.2

Cancel the common factors.

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Step 2.1.2.2.7.2.1

Factor $5$ out of $5$ .

$\frac{5 (- 2 x^{4})}{5 (1)} + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.7.2.2

Cancel the common factor.

$\frac{5 (- 2 x^{4})}{5 \cdot 1} + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.7.2.3

Rewrite the expression.

$\frac{- 2 x^{4}}{1} + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.2.7.2.4

Divide $- 2 x^{4}$ by $1$ .

$- 2 x^{4} + \frac{d}{d x} [24 x^{3}]$

Step 2.1.2.3

Evaluate $\frac{d}{d x} [24 x^{3}]$ .

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Step 2.1.2.3.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{3}$ with respect to $x$ is $24 \frac{d}{d x} [x^{3}]$ .

$- 2 x^{4} + 24 \frac{d}{d x} [x^{3}]$

Step 2.1.2.3.2

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

$- 2 x^{4} + 24 (3 x^{2})$

Step 2.1.2.3.3

Multiply $3$ by $24$ .

$f'' (x) = - 2 x^{4} + 72 x^{2}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 2 x^{4} + 72 x^{2}$ .

$- 2 x^{4} + 72 x^{2}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $- 2 x^{4} + 72 x^{2} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$- 2 x^{4} + 72 x^{2} = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- 2 {(x^{2})}^{2} + 72 x^{2} = 0$

Step 2.2.2.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 2 u^{2} + 72 u = 0$

Step 2.2.2.3

Factor $2 u$ out of $- 2 u^{2} + 72 u$ .

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Step 2.2.2.3.1

Factor $2 u$ out of $- 2 u^{2}$ .

$2 u (- u) + 72 u = 0$

Step 2.2.2.3.2

Factor $2 u$ out of $72 u$ .

$2 u (- u) + 2 u (36) = 0$

Step 2.2.2.3.3

Factor $2 u$ out of $2 u (- u) + 2 u (36)$ .

$2 u (- u + 36) = 0$

Step 2.2.2.4

Replace all occurrences of $u$ with $x^{2}$ .

$2 x^{2} (- x^{2} + 36) = 0$

Step 2.2.3

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

$x^{2} = 0$

$- x^{2} + 36 = 0$

Step 2.2.4

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

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Step 2.2.4.1

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

$x^{2} = 0$

Step 2.2.4.2

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

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Step 2.2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.4.2.2.1

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

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

Step 2.2.4.2.2.2

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

$x = \pm 0$

Step 2.2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.2.5

Set $- x^{2} + 36$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $- x^{2} + 36$ equal to $0$ .

$- x^{2} + 36 = 0$

Step 2.2.5.2

Solve $- x^{2} + 36 = 0$ for $x$ .

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Step 2.2.5.2.1

Subtract $36$ from both sides of the equation.

$- x^{2} = - 36$

Step 2.2.5.2.2

Divide each term in $- x^{2} = - 36$ by $- 1$ and simplify.

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Step 2.2.5.2.2.1

Divide each term in $- x^{2} = - 36$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 36}{- 1}$

Step 2.2.5.2.2.2

Simplify the left side.

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Step 2.2.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 36}{- 1}$

Step 2.2.5.2.2.2.2

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

$x^{2} = \frac{- 36}{- 1}$

Step 2.2.5.2.2.3

Simplify the right side.

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Step 2.2.5.2.2.3.1

Divide $- 36$ by $- 1$ .

$x^{2} = 36$

Step 2.2.5.2.3

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

$x = \pm \sqrt{36}$

Step 2.2.5.2.4

Simplify $\pm \sqrt{36}$ .

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Step 2.2.5.2.4.1

Rewrite $36$ as $6^{2}$ .

$x = \pm \sqrt{6^{2}}$

Step 2.2.5.2.4.2

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

$x = \pm 6$

Step 2.2.5.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.5.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 6$

Step 2.2.5.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 6$

Step 2.2.5.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 6, - 6$

Step 2.2.6

The final solution is all the values that make $2 x^{2} (- x^{2} + 36) = 0$ true.

$x = 0, 6, - 6$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 6) \cup (- 6, 0) \cup (0, 6) \cup (6, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 6)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 8$ in the expression.

$f'' (- 8) = - 2 {(- 8)}^{4} + 72 {(- 8)}^{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 8$ to the power of $4$ .

$f'' (- 8) = - 2 \cdot 4096 + 72 {(- 8)}^{2}$

Step 5.2.1.2

Multiply $- 2$ by $4096$ .

$f'' (- 8) = - 8192 + 72 {(- 8)}^{2}$

Step 5.2.1.3

Raise $- 8$ to the power of $2$ .

$f'' (- 8) = - 8192 + 72 \cdot 64$

Step 5.2.1.4

Multiply $72$ by $64$ .

$f'' (- 8) = - 8192 + 4608$

Step 5.2.2

Add $- 8192$ and $4608$ .

$f'' (- 8) = - 3584$

Step 5.2.3

The final answer is $- 3584$ .

$- 3584$

Step 5.3

The graph is concave down on the interval $(- \infty, - 6)$ because $f'' (- 8)$ is negative.

Concave down on $(- \infty, - 6)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(- 6, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (- 3) = - 2 {(- 3)}^{4} + 72 {(- 3)}^{2}$

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

$f'' (- 3) = - 2 \cdot 81 + 72 {(- 3)}^{2}$

Step 6.2.1.2

Multiply $- 2$ by $81$ .

$f'' (- 3) = - 162 + 72 {(- 3)}^{2}$

Step 6.2.1.3

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

$f'' (- 3) = - 162 + 72 \cdot 9$

Step 6.2.1.4

Multiply $72$ by $9$ .

$f'' (- 3) = - 162 + 648$

Step 6.2.2

Add $- 162$ and $648$ .

$f'' (- 3) = 486$

Step 6.2.3

The final answer is $486$ .

$486$

Step 6.3

The graph is concave up on the interval $(- 6, 0)$ because $f'' (- 3)$ is positive.

Concave up on $(- 6, 0)$ since $f'' (x)$ is positive

Step 7

Substitute any number from the interval $(0, 6)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

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

$f'' (3) = - 2 {(3)}^{4} + 72 {(3)}^{2}$

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

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

$f'' (3) = - 2 \cdot 81 + 72 {(3)}^{2}$

Step 7.2.1.2

Multiply $- 2$ by $81$ .

$f'' (3) = - 162 + 72 {(3)}^{2}$

Step 7.2.1.3

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

$f'' (3) = - 162 + 72 \cdot 9$

Step 7.2.1.4

Multiply $72$ by $9$ .

$f'' (3) = - 162 + 648$

Step 7.2.2

Add $- 162$ and $648$ .

$f'' (3) = 486$

Step 7.2.3

The final answer is $486$ .

$486$

Step 7.3

The graph is concave up on the interval $(0, 6)$ because $f'' (3)$ is positive.

Concave up on $(0, 6)$ since $f'' (x)$ is positive

Step 8

Substitute any number from the interval $(6, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 8.1

Replace the variable $x$ with $8$ in the expression.

$f'' (8) = - 2 {(8)}^{4} + 72 {(8)}^{2}$

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 $8$ to the power of $4$ .

$f'' (8) = - 2 \cdot 4096 + 72 {(8)}^{2}$

Step 8.2.1.2

Multiply $- 2$ by $4096$ .

$f'' (8) = - 8192 + 72 {(8)}^{2}$

Step 8.2.1.3

Raise $8$ to the power of $2$ .

$f'' (8) = - 8192 + 72 \cdot 64$

Step 8.2.1.4

Multiply $72$ by $64$ .

$f'' (8) = - 8192 + 4608$

Step 8.2.2

Add $- 8192$ and $4608$ .

$f'' (8) = - 3584$

Step 8.2.3

The final answer is $- 3584$ .

$- 3584$

Step 8.3

The graph is concave down on the interval $(6, \infty)$ because $f'' (8)$ is negative.

Concave down on $(6, \infty)$ since $f'' (x)$ is negative

Step 9

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, - 6)$ since $f'' (x)$ is negative

Concave up on $(- 6, 0)$ since $f'' (x)$ is positive

Concave up on $(0, 6)$ since $f'' (x)$ is positive

Concave down on $(6, \infty)$ since $f'' (x)$ is negative

Step 10