Calculus Examples

$f (x) = - x^{3} + 30 x^{2} + 63 x$

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 $- x^{3} + 30 x^{2} + 63 x$ with respect to $x$ is $\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [30 x^{2}] + \frac{d}{d x} [63 x]$ .

$\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [30 x^{2}] + \frac{d}{d x} [63 x]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$- \frac{d}{d x} [x^{3}] + \frac{d}{d x} [30 x^{2}] + \frac{d}{d x} [63 x]$

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 $n = 3$ .

$- (3 x^{2}) + \frac{d}{d x} [30 x^{2}] + \frac{d}{d x} [63 x]$

Step 1.1.1.2.3

Multiply $3$ by $- 1$ .

$- 3 x^{2} + \frac{d}{d x} [30 x^{2}] + \frac{d}{d x} [63 x]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [30 x^{2}]$ .

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Step 1.1.1.3.1

Since $30$ is constant with respect to $x$ , the derivative of $30 x^{2}$ with respect to $x$ is $30 \frac{d}{d x} [x^{2}]$ .

$- 3 x^{2} + 30 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [63 x]$

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 $n = 2$ .

$- 3 x^{2} + 30 (2 x) + \frac{d}{d x} [63 x]$

Step 1.1.1.3.3

Multiply $2$ by $30$ .

$- 3 x^{2} + 60 x + \frac{d}{d x} [63 x]$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [63 x]$ .

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Step 1.1.1.4.1

Since $63$ is constant with respect to $x$ , the derivative of $63 x$ with respect to $x$ is $63 \frac{d}{d x} [x]$ .

$- 3 x^{2} + 60 x + 63 \frac{d}{d x} [x]$

Step 1.1.1.4.2

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

$- 3 x^{2} + 60 x + 63 \cdot 1$

Step 1.1.1.4.3

Multiply $63$ by $1$ .

$f' (x) = - 3 x^{2} + 60 x + 63$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $- 3 x^{2} + 60 x + 63$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [60 x] + \frac{d}{d x} [63]$ .

$\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [60 x] + \frac{d}{d x} [63]$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 1.1.2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$- 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [60 x] + \frac{d}{d x} [63]$

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 $n = 2$ .

$- 3 (2 x) + \frac{d}{d x} [60 x] + \frac{d}{d x} [63]$

Step 1.1.2.2.3

Multiply $2$ by $- 3$ .

$- 6 x + \frac{d}{d x} [60 x] + \frac{d}{d x} [63]$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [60 x]$ .

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Step 1.1.2.3.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x$ with respect to $x$ is $60 \frac{d}{d x} [x]$ .

$- 6 x + 60 \frac{d}{d x} [x] + \frac{d}{d x} [63]$

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 $n = 1$ .

$- 6 x + 60 \cdot 1 + \frac{d}{d x} [63]$

Step 1.1.2.3.3

Multiply $60$ by $1$ .

$- 6 x + 60 + \frac{d}{d x} [63]$

Step 1.1.2.4

Differentiate using the Constant Rule.

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Step 1.1.2.4.1

Since $63$ is constant with respect to $x$ , the derivative of $63$ with respect to $x$ is $0$ .

$- 6 x + 60 + 0$

Step 1.1.2.4.2

Add $- 6 x + 60$ and $0$ .

$f'' (x) = - 6 x + 60$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 6 x + 60$ .

$- 6 x + 60$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- 6 x + 60 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- 6 x + 60 = 0$

Step 1.2.2

Subtract $60$ from both sides of the equation.

$- 6 x = - 60$

Step 1.2.3

Divide each term in $- 6 x = - 60$ by $- 6$ and simplify.

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Step 1.2.3.1

Divide each term in $- 6 x = - 60$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 60}{- 6}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $- 6$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 60}{- 6}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 60}{- 6}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $- 60$ by $- 6$ .

$x = 10$

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.

$(- \infty, 10) \cup (10, \infty)$

Step 4

Substitute any number from the interval $(- \infty, 10)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (0) = - 6 \cdot 0 + 60$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $- 6$ by $0$ .

$f'' (0) = 0 + 60$

Step 4.2.2

Add $0$ and $60$ .

$f'' (0) = 60$

Step 4.2.3

The final answer is $60$ .

$60$

Step 4.3

The graph is concave up on the interval $(- \infty, 10)$ because $f'' (0)$ is positive.

Concave up on $(- \infty, 10)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(10, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $12$ in the expression.

$f'' (12) = - 6 \cdot 12 + 60$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $- 6$ by $12$ .

$f'' (12) = - 72 + 60$

Step 5.2.2

Add $- 72$ and $60$ .

$f'' (12) = - 12$

Step 5.2.3

The final answer is $- 12$ .

$- 12$

Step 5.3

The graph is concave down on the interval $(10, \infty)$ because $f'' (12)$ is negative.

Concave down on $(10, \infty)$ 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 $(- \infty, 10)$ since $f'' (x)$ is positive

Concave down on $(10, \infty)$ since $f'' (x)$ is negative

Step 7