Calculus Examples

$f (x) = \frac{1}{4} x^{4} + 4 x^{3} + 24 x^{2}$

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 $\frac{1}{4} x^{4} + 4 x^{3} + 24 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [24 x^{2}]$ .

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

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{4}]$ .

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Step 1.1.1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

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

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 $n = 4$ .

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

Step 1.1.1.2.3

Combine $4$ and $\frac{1}{4}$ .

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

Step 1.1.1.2.4

Combine $\frac{4}{4}$ and $x^{3}$ .

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

Step 1.1.1.2.5

Cancel the common factor of $4$ .

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Step 1.1.1.2.5.1

Cancel the common factor.

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

Step 1.1.1.2.5.2

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

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

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 $n = 3$ .

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

Step 1.1.1.3.3

Multiply $3$ by $4$ .

$x^{3} + 12 x^{2} + \frac{d}{d x} [24 x^{2}]$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [24 x^{2}]$ .

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Step 1.1.1.4.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{2}$ with respect to $x$ is $24 \frac{d}{d x} [x^{2}]$ .

$x^{3} + 12 x^{2} + 24 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$x^{3} + 12 x^{2} + 24 (2 x)$

Step 1.1.1.4.3

Multiply $2$ by $24$ .

$f' (x) = x^{3} + 12 x^{2} + 48 x$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Differentiate.

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Step 1.1.2.1.1

By the Sum Rule, the derivative of $x^{3} + 12 x^{2} + 48 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [48 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [48 x]$

Step 1.1.2.1.2

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

$3 x^{2} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [48 x]$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 1.1.2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} + 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [48 x]$

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 $n = 2$ .

$3 x^{2} + 12 (2 x) + \frac{d}{d x} [48 x]$

Step 1.1.2.2.3

Multiply $2$ by $12$ .

$3 x^{2} + 24 x + \frac{d}{d x} [48 x]$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [48 x]$ .

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Step 1.1.2.3.1

Since $48$ is constant with respect to $x$ , the derivative of $48 x$ with respect to $x$ is $48 \frac{d}{d x} [x]$ .

$3 x^{2} + 24 x + 48 \frac{d}{d x} [x]$

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 $n = 1$ .

$3 x^{2} + 24 x + 48 \cdot 1$

Step 1.1.2.3.3

Multiply $48$ by $1$ .

$f'' (x) = 3 x^{2} + 24 x + 48$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 24 x + 48$ .

$3 x^{2} + 24 x + 48$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $3 x^{2} + 24 x + 48 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$3 x^{2} + 24 x + 48 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $3$ out of $3 x^{2} + 24 x + 48$ .

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Step 1.2.2.1.1

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

$3 (x^{2}) + 24 x + 48 = 0$

Step 1.2.2.1.2

Factor $3$ out of $24 x$ .

$3 (x^{2}) + 3 (8 x) + 48 = 0$

Step 1.2.2.1.3

Factor $3$ out of $48$ .

$3 x^{2} + 3 (8 x) + 3 \cdot 16 = 0$

Step 1.2.2.1.4

Factor $3$ out of $3 x^{2} + 3 (8 x)$ .

$3 (x^{2} + 8 x) + 3 \cdot 16 = 0$

Step 1.2.2.1.5

Factor $3$ out of $3 (x^{2} + 8 x) + 3 \cdot 16$ .

$3 (x^{2} + 8 x + 16) = 0$

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

Rewrite $16$ as $4^{2}$ .

$3 (x^{2} + 8 x + 4^{2}) = 0$

Step 1.2.2.2.2

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

$8 x = 2 \cdot x \cdot 4$

Step 1.2.2.2.3

Rewrite the polynomial.

$3 (x^{2} + 2 \cdot x \cdot 4 + 4^{2}) = 0$

Step 1.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$3 {(x + 4)}^{2} = 0$

Step 1.2.3

Divide each term in $3 {(x + 4)}^{2} = 0$ by $3$ and simplify.

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Step 1.2.3.1

Divide each term in $3 {(x + 4)}^{2} = 0$ by $3$ .

$\frac{3 {(x + 4)}^{2}}{3} = \frac{0}{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{3 {(x + 4)}^{2}}{3} = \frac{0}{3}$

Step 1.2.3.2.1.2

Divide ${(x + 4)}^{2}$ by $1$ .

${(x + 4)}^{2} = \frac{0}{3}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $3$ .

${(x + 4)}^{2} = 0$

Step 1.2.4

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 1.2.5

Subtract $4$ from both sides of the equation.

$x = - 4$

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.

$(- \infty, - 4) \cup (- 4, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 4)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $- 6$ in the expression.

$f'' (- 6) = 3 {(- 6)}^{2} + 24 (- 6) + 48$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $- 6$ to the power of $2$ .

$f'' (- 6) = 3 \cdot 36 + 24 (- 6) + 48$

Step 4.2.1.2

Multiply $3$ by $36$ .

$f'' (- 6) = 108 + 24 (- 6) + 48$

Step 4.2.1.3

Multiply $24$ by $- 6$ .

$f'' (- 6) = 108 - 144 + 48$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $144$ from $108$ .

$f'' (- 6) = - 36 + 48$

Step 4.2.2.2

Add $- 36$ and $48$ .

$f'' (- 6) = 12$

Step 4.2.3

The final answer is $12$ .

$12$

Step 4.3

The graph is concave up on the interval $(- \infty, - 4)$ because $f'' (- 6)$ is positive.

Concave up on $(- \infty, - 4)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- 4, \infty)$ 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.

$f'' (0) = 3 {(0)}^{2} + 24 (0) + 48$

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

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

$f'' (0) = 3 \cdot 0 + 24 (0) + 48$

Step 5.2.1.2

Multiply $3$ by $0$ .

$f'' (0) = 0 + 24 (0) + 48$

Step 5.2.1.3

Multiply $24$ by $0$ .

$f'' (0) = 0 + 0 + 48$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $0$ and $0$ .

$f'' (0) = 0 + 48$

Step 5.2.2.2

Add $0$ and $48$ .

$f'' (0) = 48$

Step 5.2.3

The final answer is $48$ .

$48$

Step 5.3

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

Concave up on $(- 4, \infty)$ since $f'' (x)$ is positive

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 $(- \infty, - 4)$ since $f'' (x)$ is positive

Concave up on $(- 4, \infty)$ since $f'' (x)$ is positive

Step 7