Calculus Examples

$f (x) = x {(x + 5)}^{2}$

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

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$f' (x) = \frac{d}{d x} (x ((x + 5) (x + 5)))$

Step 1.1.1.2

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 1.1.1.2.1

Apply the distributive property.

$f' (x) = \frac{d}{d x} (x (x (x + 5) + 5 (x + 5)))$

Step 1.1.1.2.2

Apply the distributive property.

$f' (x) = \frac{d}{d x} (x (x \cdot x + x \cdot 5 + 5 (x + 5)))$

Step 1.1.1.2.3

Apply the distributive property.

$f' (x) = \frac{d}{d x} (x (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5))$

Step 1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.3.1

Simplify each term.

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Step 1.1.1.3.1.1

Multiply $x$ by $x$ .

$f' (x) = \frac{d}{d x} (x (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5))$

Step 1.1.1.3.1.2

Move $5$ to the left of $x$ .

$f' (x) = \frac{d}{d x} (x (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5))$

Step 1.1.1.3.1.3

Multiply $5$ by $5$ .

$f' (x) = \frac{d}{d x} (x (x^{2} + 5 x + 5 x + 25))$

Step 1.1.1.3.2

Add $5 x$ and $5 x$ .

$f' (x) = \frac{d}{d x} (x (x^{2} + 10 x + 25))$

Step 1.1.1.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{2} + 10 x + 25$ .

$f' (x) = x \frac{d}{d x} (x^{2} + 10 x + 25) + (x^{2} + 10 x + 25) \frac{d}{d x} (x)$

Step 1.1.1.5

Differentiate.

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Step 1.1.1.5.1

By the Sum Rule, the derivative of $x^{2} + 10 x + 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [25]$ .

$f' (x) = x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (10 x) + \frac{d}{d x} (25)) + (x^{2} + 10 x + 25) \frac{d}{d x} (x)$

Step 1.1.1.5.2

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

$f' (x) = x (2 x + \frac{d}{d x} (10 x) + \frac{d}{d x} (25)) + (x^{2} + 10 x + 25) \frac{d}{d x} (x)$

Step 1.1.1.5.3

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$f' (x) = x (2 x + 10 \frac{d}{d x} (x) + \frac{d}{d x} (25)) + (x^{2} + 10 x + 25) \frac{d}{d x} (x)$

Step 1.1.1.5.4

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

$f' (x) = x (2 x + 10 \cdot 1 + \frac{d}{d x} (25)) + (x^{2} + 10 x + 25) \frac{d}{d x} (x)$

Step 1.1.1.5.5

Multiply $10$ by $1$ .

$f' (x) = x (2 x + 10 + \frac{d}{d x} (25)) + (x^{2} + 10 x + 25) \frac{d}{d x} (x)$

Step 1.1.1.5.6

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$f' (x) = x (2 x + 10 + 0) + (x^{2} + 10 x + 25) \frac{d}{d x} (x)$

Step 1.1.1.5.7

Add $2 x + 10$ and $0$ .

$f' (x) = x (2 x + 10) + (x^{2} + 10 x + 25) \frac{d}{d x} (x)$

Step 1.1.1.5.8

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

$f' (x) = x (2 x + 10) + (x^{2} + 10 x + 25) \cdot 1$

Step 1.1.1.5.9

Multiply $x^{2} + 10 x + 25$ by $1$ .

$f' (x) = x (2 x + 10) + x^{2} + 10 x + 25$

Step 1.1.1.6

Simplify.

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Step 1.1.1.6.1

Apply the distributive property.

$f' (x) = x (2 x) + x \cdot 10 + x^{2} + 10 x + 25$

Step 1.1.1.6.2

Combine terms.

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Step 1.1.1.6.2.1

Raise $x$ to the power of $1$ .

$f' (x) = 2 (x \cdot x) + x \cdot 10 + x^{2} + 10 x + 25$

Step 1.1.1.6.2.2

Raise $x$ to the power of $1$ .

$f' (x) = 2 (x \cdot x) + x \cdot 10 + x^{2} + 10 x + 25$

Step 1.1.1.6.2.3

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

$f' (x) = 2 x^{1 + 1} + x \cdot 10 + x^{2} + 10 x + 25$

Step 1.1.1.6.2.4

Add $1$ and $1$ .

$f' (x) = 2 x^{2} + x \cdot 10 + x^{2} + 10 x + 25$

Step 1.1.1.6.2.5

Move $10$ to the left of $x$ .

$f' (x) = 2 x^{2} + 10 \cdot x + x^{2} + 10 x + 25$

Step 1.1.1.6.2.6

Add $2 x^{2}$ and $x^{2}$ .

$f' (x) = 3 x^{2} + 10 x + 10 x + 25$

Step 1.1.1.6.2.7

Add $10 x$ and $10 x$ .

$f' (x) = 3 x^{2} + 20 x + 25$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $3 x^{2} + 20 x + 25$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [20 x] + \frac{d}{d x} [25]$ .

$f'' (x) = \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (20 x) + \frac{d}{d x} (25)$

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

$f'' (x) = 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (20 x) + \frac{d}{d x} (25)$

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 $n = 2$ .

$f'' (x) = 3 (2 x) + \frac{d}{d x} (20 x) + \frac{d}{d x} (25)$

Step 1.1.2.2.3

Multiply $2$ by $3$ .

$f'' (x) = 6 x + \frac{d}{d x} (20 x) + \frac{d}{d x} (25)$

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$f'' (x) = 6 x + 20 \frac{d}{d x} (x) + \frac{d}{d x} (25)$

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 $n = 1$ .

$f'' (x) = 6 x + 20 \cdot 1 + \frac{d}{d x} (25)$

Step 1.1.2.3.3

Multiply $20$ by $1$ .

$f'' (x) = 6 x + 20 + \frac{d}{d x} (25)$

Step 1.1.2.4

Differentiate using the Constant Rule.

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Step 1.1.2.4.1

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$f'' (x) = 6 x + 20 + 0$

Step 1.1.2.4.2

Add $6 x + 20$ and $0$ .

$f'' (x) = 6 x + 20$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $6 x + 20$ .

$6 x + 20$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $6 x + 20 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$6 x + 20 = 0$

Step 1.2.2

Subtract $20$ from both sides of the equation.

$6 x = - 20$

Step 1.2.3

Divide each term in $6 x = - 20$ by $6$ and simplify.

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Step 1.2.3.1

Divide each term in $6 x = - 20$ by $6$ .

$\frac{6 x}{6} = \frac{- 20}{6}$

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.

$\frac{6 x}{6} = \frac{- 20}{6}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 20}{6}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Cancel the common factor of $- 20$ and $6$ .

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Step 1.2.3.3.1.1

Factor $2$ out of $- 20$ .

$x = \frac{2 (- 10)}{6}$

Step 1.2.3.3.1.2

Cancel the common factors.

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Step 1.2.3.3.1.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot - 10}{2 \cdot 3}$

Step 1.2.3.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 10}{2 \cdot 3}$

Step 1.2.3.3.1.2.3

Rewrite the expression.

$x = \frac{- 10}{3}$

Step 1.2.3.3.2

Move the negative in front of the fraction.

$x = - \frac{10}{3}$

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.

$(- \infty, - \frac{10}{3}) \cup (- \frac{10}{3}, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \frac{10}{3})$ 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.

$f'' (- 6) = 6 (- 6) + 20$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $6$ by $- 6$ .

$f'' (- 6) = - 36 + 20$

Step 4.2.2

Add $- 36$ and $20$ .

$f'' (- 6) = - 16$

Step 4.2.3

The final answer is $- 16$ .

$- 16$

Step 4.3

The graph is concave down on the interval $(- \infty, - \frac{10}{3})$ because $f'' (- 6)$ is negative.

Concave down on $(- \infty, - \frac{10}{3})$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- \frac{10}{3}, \infty)$ 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.

$f'' (0) = 6 (0) + 20$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $6$ by $0$ .

$f'' (0) = 0 + 20$

Step 5.2.2

Add $0$ and $20$ .

$f'' (0) = 20$

Step 5.2.3

The final answer is $20$ .

$20$

Step 5.3

The graph is concave up on the interval $(- \frac{10}{3}, \infty)$ because $f'' (0)$ is positive.

Concave up on $(- \frac{10}{3}, \infty)$ 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 down on $(- \infty, - \frac{10}{3})$ since $f'' (x)$ is negative

Concave up on $(- \frac{10}{3}, \infty)$ since $f'' (x)$ is positive

Step 7