Calculus Examples

$- \frac{3}{5} x^{5} + 7 x^{4}$

Step 1

Write $- \frac{3}{5} x^{5} + 7 x^{4}$ as a function.

$f (x) = - \frac{3}{5} x^{5} + 7 x^{4}$

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 $- \frac{3}{5} x^{5} + 7 x^{4}$ with respect to $x$ is $\frac{d}{d x} [- \frac{3}{5} x^{5}] + \frac{d}{d x} [7 x^{4}]$ .

$\frac{d}{d x} [- \frac{3}{5} x^{5}] + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

$- \frac{3}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [7 x^{4}]$

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 $n = 5$ .

$- \frac{3}{5} (5 x^{4}) + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2.3

Multiply $5$ by $- 1$ .

$- 5 (\frac{3}{5}) x^{4} + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2.4

Combine $- 5$ and $\frac{3}{5}$ .

$\frac{- 5 \cdot 3}{5} x^{4} + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2.5

Multiply $- 5$ by $3$ .

$\frac{- 15}{5} x^{4} + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2.6

Combine $\frac{- 15}{5}$ and $x^{4}$ .

$\frac{- 15 x^{4}}{5} + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2.7

Cancel the common factor of $- 15$ and $5$ .

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Step 2.1.1.2.7.1

Factor $5$ out of $- 15 x^{4}$ .

$\frac{5 (- 3 x^{4})}{5} + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2.7.2

Cancel the common factors.

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Step 2.1.1.2.7.2.1

Factor $5$ out of $5$ .

$\frac{5 (- 3 x^{4})}{5 (1)} + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2.7.2.2

Cancel the common factor.

$\frac{5 (- 3 x^{4})}{5 \cdot 1} + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2.7.2.3

Rewrite the expression.

$\frac{- 3 x^{4}}{1} + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.2.7.2.4

Divide $- 3 x^{4}$ by $1$ .

$- 3 x^{4} + \frac{d}{d x} [7 x^{4}]$

Step 2.1.1.3

Evaluate $\frac{d}{d x} [7 x^{4}]$ .

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Step 2.1.1.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{4}$ with respect to $x$ is $7 \frac{d}{d x} [x^{4}]$ .

$- 3 x^{4} + 7 \frac{d}{d x} [x^{4}]$

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 $n = 4$ .

$- 3 x^{4} + 7 (4 x^{3})$

Step 2.1.1.3.3

Multiply $4$ by $7$ .

$f' (x) = - 3 x^{4} + 28 x^{3}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $- 3 x^{4} + 28 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [28 x^{3}]$ .

$\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [28 x^{3}]$

Step 2.1.2.2

Evaluate $\frac{d}{d x} [- 3 x^{4}]$ .

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Step 2.1.2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{4}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{4}]$ .

$- 3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [28 x^{3}]$

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 $n = 4$ .

$- 3 (4 x^{3}) + \frac{d}{d x} [28 x^{3}]$

Step 2.1.2.2.3

Multiply $4$ by $- 3$ .

$- 12 x^{3} + \frac{d}{d x} [28 x^{3}]$

Step 2.1.2.3

Evaluate $\frac{d}{d x} [28 x^{3}]$ .

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Step 2.1.2.3.1

Since $28$ is constant with respect to $x$ , the derivative of $28 x^{3}$ with respect to $x$ is $28 \frac{d}{d x} [x^{3}]$ .

$- 12 x^{3} + 28 \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

$- 12 x^{3} + 28 (3 x^{2})$

Step 2.1.2.3.3

Multiply $3$ by $28$ .

$f'' (x) = - 12 x^{3} + 84 x^{2}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 12 x^{3} + 84 x^{2}$ .

$- 12 x^{3} + 84 x^{2}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $- 12 x^{3} + 84 x^{2} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$- 12 x^{3} + 84 x^{2} = 0$

Step 2.2.2

Factor $- 12 x^{2}$ out of $- 12 x^{3} + 84 x^{2}$ .

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Step 2.2.2.1

Factor $- 12 x^{2}$ out of $- 12 x^{3}$ .

$- 12 x^{2} x + 84 x^{2} = 0$

Step 2.2.2.2

Factor $- 12 x^{2}$ out of $84 x^{2}$ .

$- 12 x^{2} x - 12 x^{2} \cdot - 7 = 0$

Step 2.2.2.3

Factor $- 12 x^{2}$ out of $- 12 x^{2} (x) - 12 x^{2} (- 7)$ .

$- 12 x^{2} (x - 7) = 0$

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 $0$ .

$x^{2} = 0$

$x - 7 = 0$

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 $x - 7$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 2.2.5.2

Add $7$ to both sides of the equation.

$x = 7$

Step 2.2.6

The final solution is all the values that make $- 12 x^{2} (x - 7) = 0$ true.

$x = 0, 7$

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.

$(- \infty, 0) \cup (0, 7) \cup (7, \infty)$

Step 5

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

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Step 5.1

Replace the variable $x$ with $- 2$ in the expression.

$f'' (- 2) = - 12 {(- 2)}^{3} + 84 {(- 2)}^{2}$

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

$f'' (- 2) = - 12 \cdot - 8 + 84 {(- 2)}^{2}$

Step 5.2.1.2

Multiply $- 12$ by $- 8$ .

$f'' (- 2) = 96 + 84 {(- 2)}^{2}$

Step 5.2.1.3

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

$f'' (- 2) = 96 + 84 \cdot 4$

Step 5.2.1.4

Multiply $84$ by $4$ .

$f'' (- 2) = 96 + 336$

Step 5.2.2

Add $96$ and $336$ .

$f'' (- 2) = 432$

Step 5.2.3

The final answer is $432$ .

$432$

Step 5.3

The graph is concave up on the interval $(- \infty, 0)$ because $f'' (- 2)$ is positive.

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

Step 6

Substitute any number from the interval $(0, 7)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $4$ in the expression.

$f'' (4) = - 12 {(4)}^{3} + 84 {(4)}^{2}$

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 $4$ to the power of $3$ .

$f'' (4) = - 12 \cdot 64 + 84 {(4)}^{2}$

Step 6.2.1.2

Multiply $- 12$ by $64$ .

$f'' (4) = - 768 + 84 {(4)}^{2}$

Step 6.2.1.3

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

$f'' (4) = - 768 + 84 \cdot 16$

Step 6.2.1.4

Multiply $84$ by $16$ .

$f'' (4) = - 768 + 1344$

Step 6.2.2

Add $- 768$ and $1344$ .

$f'' (4) = 576$

Step 6.2.3

The final answer is $576$ .

$576$

Step 6.3

The graph is concave up on the interval $(0, 7)$ because $f'' (4)$ is positive.

Concave up on $(0, 7)$ since $f'' (x)$ is positive

Step 7

Substitute any number from the interval $(7, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $10$ in the expression.

$f'' (10) = - 12 {(10)}^{3} + 84 {(10)}^{2}$

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 $10$ to the power of $3$ .

$f'' (10) = - 12 \cdot 1000 + 84 {(10)}^{2}$

Step 7.2.1.2

Multiply $- 12$ by $1000$ .

$f'' (10) = - 12000 + 84 {(10)}^{2}$

Step 7.2.1.3

Raise $10$ to the power of $2$ .

$f'' (10) = - 12000 + 84 \cdot 100$

Step 7.2.1.4

Multiply $84$ by $100$ .

$f'' (10) = - 12000 + 8400$

Step 7.2.2

Add $- 12000$ and $8400$ .

$f'' (10) = - 3600$

Step 7.2.3

The final answer is $- 3600$ .

$- 3600$

Step 7.3

The graph is concave down on the interval $(7, \infty)$ because $f'' (10)$ is negative.

Concave down on $(7, \infty)$ since $f'' (x)$ is negative

Step 8

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

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

Concave up on $(0, 7)$ since $f'' (x)$ is positive

Concave down on $(7, \infty)$ since $f'' (x)$ is negative

Step 9