Calculus Examples

$f (x) = - \frac{1}{5} x^{6} + 5 x^{4}$

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 $- \frac{1}{5} x^{6} + 5 x^{4}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{5} x^{6}] + \frac{d}{d x} [5 x^{4}]$ .

$\frac{d}{d x} [- \frac{1}{5} x^{6}] + \frac{d}{d x} [5 x^{4}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [- \frac{1}{5} x^{6}]$ .

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Step 1.1.1.2.1

Since $- \frac{1}{5}$ is constant with respect to $x$ , the derivative of $- \frac{1}{5} x^{6}$ with respect to $x$ is $- \frac{1}{5} \frac{d}{d x} [x^{6}]$ .

$- \frac{1}{5} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [5 x^{4}]$

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 $n = 6$ .

$- \frac{1}{5} (6 x^{5}) + \frac{d}{d x} [5 x^{4}]$

Step 1.1.1.2.3

Multiply $6$ by $- 1$ .

$- 6 (\frac{1}{5}) x^{5} + \frac{d}{d x} [5 x^{4}]$

Step 1.1.1.2.4

Combine $- 6$ and $\frac{1}{5}$ .

$\frac{- 6}{5} x^{5} + \frac{d}{d x} [5 x^{4}]$

Step 1.1.1.2.5

Combine $\frac{- 6}{5}$ and $x^{5}$ .

$\frac{- 6 x^{5}}{5} + \frac{d}{d x} [5 x^{4}]$

Step 1.1.1.2.6

Move the negative in front of the fraction.

$- \frac{6 x^{5}}{5} + \frac{d}{d x} [5 x^{4}]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 1.1.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$- \frac{6 x^{5}}{5} + 5 \frac{d}{d x} [x^{4}]$

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 $n = 4$ .

$- \frac{6 x^{5}}{5} + 5 (4 x^{3})$

Step 1.1.1.3.3

Multiply $4$ by $5$ .

$f' (x) = - \frac{6 x^{5}}{5} + 20 x^{3}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $- \frac{6 x^{5}}{5} + 20 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- \frac{6 x^{5}}{5}] + \frac{d}{d x} [20 x^{3}]$ .

$\frac{d}{d x} [- \frac{6 x^{5}}{5}] + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [- \frac{6 x^{5}}{5}]$ .

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Step 1.1.2.2.1

Since $- \frac{6}{5}$ is constant with respect to $x$ , the derivative of $- \frac{6 x^{5}}{5}$ with respect to $x$ is $- \frac{6}{5} \frac{d}{d x} [x^{5}]$ .

$- \frac{6}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [20 x^{3}]$

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 $n = 5$ .

$- \frac{6}{5} (5 x^{4}) + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2.3

Multiply $5$ by $- 1$ .

$- 5 (\frac{6}{5}) x^{4} + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2.4

Combine $- 5$ and $\frac{6}{5}$ .

$\frac{- 5 \cdot 6}{5} x^{4} + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2.5

Multiply $- 5$ by $6$ .

$\frac{- 30}{5} x^{4} + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2.6

Combine $\frac{- 30}{5}$ and $x^{4}$ .

$\frac{- 30 x^{4}}{5} + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2.7

Cancel the common factor of $- 30$ and $5$ .

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Step 1.1.2.2.7.1

Factor $5$ out of $- 30 x^{4}$ .

$\frac{5 (- 6 x^{4})}{5} + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2.7.2

Cancel the common factors.

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Step 1.1.2.2.7.2.1

Factor $5$ out of $5$ .

$\frac{5 (- 6 x^{4})}{5 (1)} + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2.7.2.2

Cancel the common factor.

$\frac{5 (- 6 x^{4})}{5 \cdot 1} + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2.7.2.3

Rewrite the expression.

$\frac{- 6 x^{4}}{1} + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.2.7.2.4

Divide $- 6 x^{4}$ by $1$ .

$- 6 x^{4} + \frac{d}{d x} [20 x^{3}]$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [20 x^{3}]$ .

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Step 1.1.2.3.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x^{3}$ with respect to $x$ is $20 \frac{d}{d x} [x^{3}]$ .

$- 6 x^{4} + 20 \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

$- 6 x^{4} + 20 (3 x^{2})$

Step 1.1.2.3.3

Multiply $3$ by $20$ .

$f'' (x) = - 6 x^{4} + 60 x^{2}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 6 x^{4} + 60 x^{2}$ .

$- 6 x^{4} + 60 x^{2}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- 6 x^{4} + 60 x^{2} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- 6 x^{4} + 60 x^{2} = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- 6 {(x^{2})}^{2} + 60 x^{2} = 0$

Step 1.2.2.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 6 u^{2} + 60 u = 0$

Step 1.2.2.3

Factor $6 u$ out of $- 6 u^{2} + 60 u$ .

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Step 1.2.2.3.1

Factor $6 u$ out of $- 6 u^{2}$ .

$6 u (- u) + 60 u = 0$

Step 1.2.2.3.2

Factor $6 u$ out of $60 u$ .

$6 u (- u) + 6 u (10) = 0$

Step 1.2.2.3.3

Factor $6 u$ out of $6 u (- u) + 6 u (10)$ .

$6 u (- u + 10) = 0$

Step 1.2.2.4

Replace all occurrences of $u$ with $x^{2}$ .

$6 x^{2} (- x^{2} + 10) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$- x^{2} + 10 = 0$

Step 1.2.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 1.2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 1.2.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 1.2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.5

Set $- x^{2} + 10$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $- x^{2} + 10$ equal to $0$ .

$- x^{2} + 10 = 0$

Step 1.2.5.2

Solve $- x^{2} + 10 = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $10$ from both sides of the equation.

$- x^{2} = - 10$

Step 1.2.5.2.2

Divide each term in $- x^{2} = - 10$ by $- 1$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $- x^{2} = - 10$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 10}{- 1}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 10}{- 1}$

Step 1.2.5.2.2.2.2

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

$x^{2} = \frac{- 10}{- 1}$

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Divide $- 10$ by $- 1$ .

$x^{2} = 10$

Step 1.2.5.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{10}$

Step 1.2.5.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{10}$

Step 1.2.5.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{10}$

Step 1.2.5.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{10}, - \sqrt{10}$

Step 1.2.6

The final solution is all the values that make $6 x^{2} (- x^{2} + 10) = 0$ true.

$x = 0, \sqrt{10}, - \sqrt{10}$

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.

$(- \infty, - \sqrt{10}) \cup (- \sqrt{10}, 0) \cup (0, \sqrt{10}) \cup (\sqrt{10}, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \sqrt{10})$ 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.

$f'' (- 6) = - 6 {(- 6)}^{4} + 60 {(- 6)}^{2}$

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

Multiply $- 6$ by ${(- 6)}^{4}$ by adding the exponents.

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Step 4.2.1.1.1

Multiply $- 6$ by ${(- 6)}^{4}$ .

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Step 4.2.1.1.1.1

Raise $- 6$ to the power of $1$ .

$f'' (- 6) = (- 6) {(- 6)}^{4} + 60 {(- 6)}^{2}$

Step 4.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (- 6) = {(- 6)}^{1 + 4} + 60 {(- 6)}^{2}$

Step 4.2.1.1.2

Add $1$ and $4$ .

$f'' (- 6) = {(- 6)}^{5} + 60 {(- 6)}^{2}$

Step 4.2.1.2

Raise $- 6$ to the power of $5$ .

$f'' (- 6) = - 7776 + 60 {(- 6)}^{2}$

Step 4.2.1.3

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

$f'' (- 6) = - 7776 + 60 \cdot 36$

Step 4.2.1.4

Multiply $60$ by $36$ .

$f'' (- 6) = - 7776 + 2160$

Step 4.2.2

Add $- 7776$ and $2160$ .

$f'' (- 6) = - 5616$

Step 4.2.3

The final answer is $- 5616$ .

$- 5616$

Step 4.3

The graph is concave down on the interval $(- \infty, - \sqrt{10})$ because $f'' (- 6)$ is negative.

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

Step 5

Substitute any number from the interval $(- \sqrt{10}, 0)$ 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.

$f'' (- 2) = - 6 {(- 2)}^{4} + 60 {(- 2)}^{2}$

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 $4$ .

$f'' (- 2) = - 6 \cdot 16 + 60 {(- 2)}^{2}$

Step 5.2.1.2

Multiply $- 6$ by $16$ .

$f'' (- 2) = - 96 + 60 {(- 2)}^{2}$

Step 5.2.1.3

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

$f'' (- 2) = - 96 + 60 \cdot 4$

Step 5.2.1.4

Multiply $60$ by $4$ .

$f'' (- 2) = - 96 + 240$

Step 5.2.2

Add $- 96$ and $240$ .

$f'' (- 2) = 144$

Step 5.2.3

The final answer is $144$ .

$144$

Step 5.3

The graph is concave up on the interval $(- \sqrt{10}, 0)$ because $f'' (- 2)$ is positive.

Concave up on $(- \sqrt{10}, 0)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(0, \sqrt{10})$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (2) = - 6 {(2)}^{4} + 60 {(2)}^{2}$

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 $2$ to the power of $4$ .

$f'' (2) = - 6 \cdot 16 + 60 {(2)}^{2}$

Step 6.2.1.2

Multiply $- 6$ by $16$ .

$f'' (2) = - 96 + 60 {(2)}^{2}$

Step 6.2.1.3

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

$f'' (2) = - 96 + 60 \cdot 4$

Step 6.2.1.4

Multiply $60$ by $4$ .

$f'' (2) = - 96 + 240$

Step 6.2.2

Add $- 96$ and $240$ .

$f'' (2) = 144$

Step 6.2.3

The final answer is $144$ .

$144$

Step 6.3

The graph is concave up on the interval $(0, \sqrt{10})$ because $f'' (2)$ is positive.

Concave up on $(0, \sqrt{10})$ since $f'' (x)$ is positive

Step 7

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

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Step 7.1

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

$f'' (6) = - 6 {(6)}^{4} + 60 {(6)}^{2}$

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 $6$ to the power of $4$ .

$f'' (6) = - 6 \cdot 1296 + 60 {(6)}^{2}$

Step 7.2.1.2

Multiply $- 6$ by $1296$ .

$f'' (6) = - 7776 + 60 {(6)}^{2}$

Step 7.2.1.3

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

$f'' (6) = - 7776 + 60 \cdot 36$

Step 7.2.1.4

Multiply $60$ by $36$ .

$f'' (6) = - 7776 + 2160$

Step 7.2.2

Add $- 7776$ and $2160$ .

$f'' (6) = - 5616$

Step 7.2.3

The final answer is $- 5616$ .

$- 5616$

Step 7.3

The graph is concave down on the interval $(\sqrt{10}, \infty)$ because $f'' (6)$ is negative.

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

Concave up on $(- \sqrt{10}, 0)$ since $f'' (x)$ is positive

Concave up on $(0, \sqrt{10})$ since $f'' (x)$ is positive

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

Step 9