Calculus Examples

$f (x) = - \frac{1}{6} x^{4} + 4 x^{3} - 36 x^{2}$

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 $- \frac{1}{6} x^{4} + 4 x^{3} - 36 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{6} x^{4}] + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$ .

$\frac{d}{d x} [- \frac{1}{6} x^{4}] + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [- \frac{1}{6} x^{4}]$ .

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Step 1.1.1.2.1

Since $- \frac{1}{6}$ is constant with respect to $x$ , the derivative of $- \frac{1}{6} x^{4}$ with respect to $x$ is $- \frac{1}{6} \frac{d}{d x} [x^{4}]$ .

$- \frac{1}{6} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

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 $n = 4$ .

$- \frac{1}{6} (4 x^{3}) + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.2.3

Multiply $4$ by $- 1$ .

$- 4 (\frac{1}{6}) x^{3} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.2.4

Combine $- 4$ and $\frac{1}{6}$ .

$\frac{- 4}{6} x^{3} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.2.5

Combine $\frac{- 4}{6}$ and $x^{3}$ .

$\frac{- 4 x^{3}}{6} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.2.6

Cancel the common factor of $- 4$ and $6$ .

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Step 1.1.1.2.6.1

Factor $2$ out of $- 4 x^{3}$ .

$\frac{2 (- 2 x^{3})}{6} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.2.6.2

Cancel the common factors.

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Step 1.1.1.2.6.2.1

Factor $2$ out of $6$ .

$\frac{2 (- 2 x^{3})}{2 (3)} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.2.6.2.2

Cancel the common factor.

$\frac{2 (- 2 x^{3})}{2 \cdot 3} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.2.6.2.3

Rewrite the expression.

$\frac{- 2 x^{3}}{3} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.2.7

Move the negative in front of the fraction.

$- \frac{2 x^{3}}{3} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

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

$- \frac{2 x^{3}}{3} + 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

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 $n = 3$ .

$- \frac{2 x^{3}}{3} + 4 (3 x^{2}) + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.3.3

Multiply $3$ by $4$ .

$- \frac{2 x^{3}}{3} + 12 x^{2} + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [- 36 x^{2}]$ .

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Step 1.1.1.4.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x^{2}$ with respect to $x$ is $- 36 \frac{d}{d x} [x^{2}]$ .

$- \frac{2 x^{3}}{3} + 12 x^{2} - 36 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$- \frac{2 x^{3}}{3} + 12 x^{2} - 36 (2 x)$

Step 1.1.1.4.3

Multiply $2$ by $- 36$ .

$f' (x) = - \frac{2 x^{3}}{3} + 12 x^{2} - 72 x$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $- \frac{2 x^{3}}{3} + 12 x^{2} - 72 x$ with respect to $x$ is $\frac{d}{d x} [- \frac{2 x^{3}}{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$ .

$\frac{d}{d x} [- \frac{2 x^{3}}{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

$- \frac{2}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

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 $n = 3$ .

$- \frac{2}{3} (3 x^{2}) + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2.3

Multiply $3$ by $- 1$ .

$- 3 (\frac{2}{3}) x^{2} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2.4

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

$\frac{- 3 \cdot 2}{3} x^{2} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2.5

Multiply $- 3$ by $2$ .

$\frac{- 6}{3} x^{2} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2.6

Combine $\frac{- 6}{3}$ and $x^{2}$ .

$\frac{- 6 x^{2}}{3} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2.7

Cancel the common factor of $- 6$ and $3$ .

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Step 1.1.2.2.7.1

Factor $3$ out of $- 6 x^{2}$ .

$\frac{3 (- 2 x^{2})}{3} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2.7.2

Cancel the common factors.

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Step 1.1.2.2.7.2.1

Factor $3$ out of $3$ .

$\frac{3 (- 2 x^{2})}{3 (1)} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2.7.2.2

Cancel the common factor.

$\frac{3 (- 2 x^{2})}{3 \cdot 1} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2.7.2.3

Rewrite the expression.

$\frac{- 2 x^{2}}{1} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.2.7.2.4

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

$- 2 x^{2} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x]$

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$- 2 x^{2} + 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 72 x]$

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 $n = 2$ .

$- 2 x^{2} + 12 (2 x) + \frac{d}{d x} [- 72 x]$

Step 1.1.2.3.3

Multiply $2$ by $12$ .

$- 2 x^{2} + 24 x + \frac{d}{d x} [- 72 x]$

Step 1.1.2.4

Evaluate $\frac{d}{d x} [- 72 x]$ .

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Step 1.1.2.4.1

Since $- 72$ is constant with respect to $x$ , the derivative of $- 72 x$ with respect to $x$ is $- 72 \frac{d}{d x} [x]$ .

$- 2 x^{2} + 24 x - 72 \frac{d}{d x} [x]$

Step 1.1.2.4.2

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

$- 2 x^{2} + 24 x - 72 \cdot 1$

Step 1.1.2.4.3

Multiply $- 72$ by $1$ .

$f'' (x) = - 2 x^{2} + 24 x - 72$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 2 x^{2} + 24 x - 72$ .

$- 2 x^{2} + 24 x - 72$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- 2 x^{2} + 24 x - 72 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- 2 x^{2} + 24 x - 72 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 2$ out of $- 2 x^{2} + 24 x - 72$ .

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Step 1.2.2.1.1

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

$- 2 x^{2} + 24 x - 72 = 0$

Step 1.2.2.1.2

Factor $- 2$ out of $24 x$ .

$- 2 x^{2} - 2 (- 12 x) - 72 = 0$

Step 1.2.2.1.3

Factor $- 2$ out of $- 72$ .

$- 2 x^{2} - 2 (- 12 x) - 2 \cdot 36 = 0$

Step 1.2.2.1.4

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

$- 2 (x^{2} - 12 x) - 2 \cdot 36 = 0$

Step 1.2.2.1.5

Factor $- 2$ out of $- 2 (x^{2} - 12 x) - 2 (36)$ .

$- 2 (x^{2} - 12 x + 36) = 0$

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

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

$- 2 (x^{2} - 12 x + 6^{2}) = 0$

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.

$12 x = 2 \cdot x \cdot 6$

Step 1.2.2.2.3

Rewrite the polynomial.

$- 2 (x^{2} - 2 \cdot x \cdot 6 + 6^{2}) = 0$

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 $b = 6$ .

$- 2 {(x - 6)}^{2} = 0$

Step 1.2.3

Divide each term in $- 2 {(x - 6)}^{2} = 0$ by $- 2$ and simplify.

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Step 1.2.3.1

Divide each term in $- 2 {(x - 6)}^{2} = 0$ by $- 2$ .

$\frac{- 2 {(x - 6)}^{2}}{- 2} = \frac{0}{- 2}$

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.

$\frac{- 2 {(x - 6)}^{2}}{- 2} = \frac{0}{- 2}$

Step 1.2.3.2.1.2

Divide ${(x - 6)}^{2}$ by $1$ .

${(x - 6)}^{2} = \frac{0}{- 2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $- 2$ .

${(x - 6)}^{2} = 0$

Step 1.2.4

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 1.2.5

Add $6$ to both sides of the equation.

$x = 6$

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.

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

Step 4

Substitute any number from the interval $(- \infty, 6)$ 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.

$f'' (0) = - 2 {(0)}^{2} + 24 (0) - 72$

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

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

$f'' (0) = - 2 \cdot 0 + 24 (0) - 72$

Step 4.2.1.2

Multiply $- 2$ by $0$ .

$f'' (0) = 0 + 24 (0) - 72$

Step 4.2.1.3

Multiply $24$ by $0$ .

$f'' (0) = 0 + 0 - 72$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $0$ and $0$ .

$f'' (0) = 0 - 72$

Step 4.2.2.2

Subtract $72$ from $0$ .

$f'' (0) = - 72$

Step 4.2.3

The final answer is $- 72$ .

$- 72$

Step 4.3

The graph is concave down on the interval $(- \infty, 6)$ because $f'' (0)$ is negative.

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

Step 5

Substitute any number from the interval $(6, \infty)$ 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.

$f'' (8) = - 2 {(8)}^{2} + 24 (8) - 72$

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 $2$ .

$f'' (8) = - 2 \cdot 64 + 24 (8) - 72$

Step 5.2.1.2

Multiply $- 2$ by $64$ .

$f'' (8) = - 128 + 24 (8) - 72$

Step 5.2.1.3

Multiply $24$ by $8$ .

$f'' (8) = - 128 + 192 - 72$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 128$ and $192$ .

$f'' (8) = 64 - 72$

Step 5.2.2.2

Subtract $72$ from $64$ .

$f'' (8) = - 8$

Step 5.2.3

The final answer is $- 8$ .

$- 8$

Step 5.3

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 6

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

Step 7