Calculus Examples

$f (x) = - \frac{1}{3} x^{3} - 7 x^{2}$

Step 1

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

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $- \frac{1}{3} x^{3} - 7 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- 7 x^{2}]$ .

$\frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$- \frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2.2

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

$- \frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2.3

Multiply $3$ by $- 1$ .

$- 3 (\frac{1}{3}) x^{2} + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2.4

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

$\frac{- 3}{3} x^{2} + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2.5

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

$\frac{- 3 x^{2}}{3} + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2.6

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

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Step 1.1.1.2.6.1

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

$\frac{3 (- x^{2})}{3} + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2.6.2

Cancel the common factors.

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Step 1.1.1.2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (- x^{2})}{3 (1)} + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2.6.2.2

Cancel the common factor.

$\frac{3 (- x^{2})}{3 \cdot 1} + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2.6.2.3

Rewrite the expression.

$\frac{- x^{2}}{1} + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.2.6.2.4

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

$- x^{2} + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 7 x^{2}]$ .

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Step 1.1.1.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{2}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{2}]$ .

$- x^{2} - 7 \frac{d}{d x} [x^{2}]$

Step 1.1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- x^{2} - 7 (2 x)$

Step 1.1.1.3.3

Multiply $2$ by $- 7$ .

$f' (x) = - x^{2} - 14 x$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $- x^{2} - 14 x$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 14 x]$ .

$\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 14 x]$

Step 1.1.2.2

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

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Step 1.1.2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$- \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 14 x]$

Step 1.1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- (2 x) + \frac{d}{d x} [- 14 x]$

Step 1.1.2.2.3

Multiply $2$ by $- 1$ .

$- 2 x + \frac{d}{d x} [- 14 x]$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 14 x]$ .

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Step 1.1.2.3.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x$ with respect to $x$ is $- 14 \frac{d}{d x} [x]$ .

$- 2 x - 14 \frac{d}{d x} [x]$

Step 1.1.2.3.2

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

$- 2 x - 14 \cdot 1$

Step 1.1.2.3.3

Multiply $- 14$ by $1$ .

$f'' (x) = - 2 x - 14$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 2 x - 14$ .

$- 2 x - 14$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- 2 x - 14 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- 2 x - 14 = 0$

Step 1.2.2

Add $14$ to both sides of the equation.

$- 2 x = 14$

Step 1.2.3

Divide each term in $- 2 x = 14$ by $- 2$ and simplify.

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Step 1.2.3.1

Divide each term in $- 2 x = 14$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{14}{- 2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{14}{- 2}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{14}{- 2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $14$ by $- 2$ .

$x = - 7$

Step 2

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 3

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

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

Step 4

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

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Step 4.1

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

$f'' (- 10) = - 2 \cdot - 10 - 14$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $- 2$ by $- 10$ .

$f'' (- 10) = 20 - 14$

Step 4.2.2

Subtract $14$ from $20$ .

$f'' (- 10) = 6$

Step 4.2.3

The final answer is $6$ .

$6$

Step 4.3

The graph is concave up on the interval $(- \infty, - 7)$ because $f'' (- 10)$ is positive.

Concave up on $(- \infty, - 7)$ since $f'' (x)$ is positive

Step 5

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

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Step 5.1

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

$f'' (0) = - 2 \cdot 0 - 14$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $- 2$ by $0$ .

$f'' (0) = 0 - 14$

Step 5.2.2

Subtract $14$ from $0$ .

$f'' (0) = - 14$

Step 5.2.3

The final answer is $- 14$ .

$- 14$

Step 5.3

The graph is concave down on the interval $(- 7, \infty)$ because $f'' (0)$ is negative.

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

Step 6

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

Concave up on $(- \infty, - 7)$ since $f'' (x)$ is positive

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

Step 7