Calculus Examples

$f (x) = 3 x {(x + 5)}^{2}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

$f' (x) = \frac{d}{d x} (3 x ((x + 5) (x + 5)))$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$f' (x) = \frac{d}{d x} (3 x (x (x + 5) + 5 (x + 5)))$

Step 1.1.2.2

Apply the distributive property.

$f' (x) = \frac{d}{d x} (3 x (x \cdot x + x \cdot 5 + 5 (x + 5)))$

Step 1.1.2.3

Apply the distributive property.

$f' (x) = \frac{d}{d x} (3 x (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5))$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$f' (x) = \frac{d}{d x} (3 x (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5))$

Step 1.1.3.1.2

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

$f' (x) = \frac{d}{d x} (3 x (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5))$

Step 1.1.3.1.3

Multiply $5$ by $5$ .

$f' (x) = \frac{d}{d x} (3 x (x^{2} + 5 x + 5 x + 25))$

Step 1.1.3.2

Add $5 x$ and $5 x$ .

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

Step 1.1.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x (x^{2} + 10 x + 25)$ with respect to $x$ is $3 \frac{d}{d x} [x (x^{2} + 10 x + 25)]$ .

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

Step 1.1.5

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 $g (x) = x^{2} + 10 x + 25$ .

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

Step 1.1.6

Differentiate.

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Step 1.1.6.1

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

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

Step 1.1.6.2

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

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

Step 1.1.6.3

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

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

Step 1.1.6.4

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

$f' (x) = 3 (x (2 x + 10 \cdot 1 + \frac{d}{d x} (25)) + (x^{2} + 10 x + 25) \frac{d}{d x} (x))$

Step 1.1.6.5

Multiply $10$ by $1$ .

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

Step 1.1.6.6

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

$f' (x) = 3 (x (2 x + 10 + 0) + (x^{2} + 10 x + 25) \frac{d}{d x} (x))$

Step 1.1.6.7

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

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

Step 1.1.6.8

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

$f' (x) = 3 (x (2 x + 10) + (x^{2} + 10 x + 25) \cdot 1)$

Step 1.1.6.9

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

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

Step 1.1.7

Simplify.

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Step 1.1.7.1

Apply the distributive property.

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

Step 1.1.7.2

Apply the distributive property.

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

Step 1.1.7.3

Combine terms.

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Step 1.1.7.3.1

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

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

Step 1.1.7.3.2

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

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

Step 1.1.7.3.3

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

$f' (x) = 3 (2 x^{1 + 1}) + 3 (x \cdot 10) + 3 x^{2} + 3 (10 x) + 3 \cdot 25$

Step 1.1.7.3.4

Add $1$ and $1$ .

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

Step 1.1.7.3.5

Multiply $2$ by $3$ .

$f' (x) = 6 x^{2} + 3 (x \cdot 10) + 3 x^{2} + 3 (10 x) + 3 \cdot 25$

Step 1.1.7.3.6

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

$f' (x) = 6 x^{2} + 3 (10 \cdot x) + 3 x^{2} + 3 (10 x) + 3 \cdot 25$

Step 1.1.7.3.7

Multiply $10$ by $3$ .

$f' (x) = 6 x^{2} + 30 x + 3 x^{2} + 3 (10 x) + 3 \cdot 25$

Step 1.1.7.3.8

Add $6 x^{2}$ and $3 x^{2}$ .

$f' (x) = 9 x^{2} + 30 x + 3 (10 x) + 3 \cdot 25$

Step 1.1.7.3.9

Multiply $10$ by $3$ .

$f' (x) = 9 x^{2} + 30 x + 30 x + 3 \cdot 25$

Step 1.1.7.3.10

Add $30 x$ and $30 x$ .

$f' (x) = 9 x^{2} + 60 x + 3 \cdot 25$

Step 1.1.7.3.11

Multiply $3$ by $25$ .

$f' (x) = 9 x^{2} + 60 x + 75$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $9 x^{2} + 60 x + 75$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [60 x] + \frac{d}{d x} [75]$ .

$f'' (x) = \frac{d}{d x} (9 x^{2}) + \frac{d}{d x} (60 x) + \frac{d}{d x} (75)$

Step 1.2.2

Evaluate $\frac{d}{d x} [9 x^{2}]$ .

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Step 1.2.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 9 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (60 x) + \frac{d}{d x} (75)$

Step 1.2.2.2

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

$f'' (x) = 9 (2 x) + \frac{d}{d x} (60 x) + \frac{d}{d x} (75)$

Step 1.2.2.3

Multiply $2$ by $9$ .

$f'' (x) = 18 x + \frac{d}{d x} (60 x) + \frac{d}{d x} (75)$

Step 1.2.3

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

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Step 1.2.3.1

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

$f'' (x) = 18 x + 60 \frac{d}{d x} (x) + \frac{d}{d x} (75)$

Step 1.2.3.2

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

$f'' (x) = 18 x + 60 \cdot 1 + \frac{d}{d x} (75)$

Step 1.2.3.3

Multiply $60$ by $1$ .

$f'' (x) = 18 x + 60 + \frac{d}{d x} (75)$

Step 1.2.4

Differentiate using the Constant Rule.

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Step 1.2.4.1

Since $75$ is constant with respect to $x$ , the derivative of $75$ with respect to $x$ is $0$ .

$f'' (x) = 18 x + 60 + 0$

Step 1.2.4.2

Add $18 x + 60$ and $0$ .

$f'' (x) = 18 x + 60$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $18 x + 60$ .

$18 x + 60$

Step 2

Set the second derivative equal to $0$ then solve the equation $18 x + 60 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$18 x + 60 = 0$

Step 2.2

Subtract $60$ from both sides of the equation.

$18 x = - 60$

Step 2.3

Divide each term in $18 x = - 60$ by $18$ and simplify.

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Step 2.3.1

Divide each term in $18 x = - 60$ by $18$ .

$\frac{18 x}{18} = \frac{- 60}{18}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $18$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{18 x}{18} = \frac{- 60}{18}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 60}{18}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $- 60$ and $18$ .

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Step 2.3.3.1.1

Factor $6$ out of $- 60$ .

$x = \frac{6 (- 10)}{18}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $6$ out of $18$ .

$x = \frac{6 \cdot - 10}{6 \cdot 3}$

Step 2.3.3.1.2.2

Cancel the common factor.

$x = \frac{6 \cdot - 10}{6 \cdot 3}$

Step 2.3.3.1.2.3

Rewrite the expression.

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

Step 2.3.3.2

Move the negative in front of the fraction.

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

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $- \frac{10}{3}$ in $f (x) = 3 x {(x + 5)}^{2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $- \frac{10}{3}$ in the expression.

$f (- \frac{10}{3}) = 3 (- \frac{10}{3}) {((- \frac{10}{3}) + 5)}^{2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.1.1

Move the leading negative in $- \frac{10}{3}$ into the numerator.

$f (- \frac{10}{3}) = 3 (\frac{- 10}{3}) \cdot {((- \frac{10}{3}) + 5)}^{2}$

Step 3.1.2.1.2

Cancel the common factor.

$f (- \frac{10}{3}) = 3 (\frac{- 10}{3}) \cdot {((- \frac{10}{3}) + 5)}^{2}$

Step 3.1.2.1.3

Rewrite the expression.

$f (- \frac{10}{3}) = - 10 {((- \frac{10}{3}) + 5)}^{2}$

Step 3.1.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{10}{3}) = - 10 {(- \frac{10}{3} + 5 \cdot \frac{3}{3})}^{2}$

Step 3.1.2.3

Combine $5$ and $\frac{3}{3}$ .

$f (- \frac{10}{3}) = - 10 {(- \frac{10}{3} + \frac{5 \cdot 3}{3})}^{2}$

Step 3.1.2.4

Combine the numerators over the common denominator.

$f (- \frac{10}{3}) = - 10 {(\frac{- 10 + 5 \cdot 3}{3})}^{2}$

Step 3.1.2.5

Simplify the numerator.

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Step 3.1.2.5.1

Multiply $5$ by $3$ .

$f (- \frac{10}{3}) = - 10 {(\frac{- 10 + 15}{3})}^{2}$

Step 3.1.2.5.2

Add $- 10$ and $15$ .

$f (- \frac{10}{3}) = - 10 {(\frac{5}{3})}^{2}$

Step 3.1.2.6

Apply the product rule to $\frac{5}{3}$ .

$f (- \frac{10}{3}) = - 10 \frac{5^{2}}{3^{2}}$

Step 3.1.2.7

Raise $5$ to the power of $2$ .

$f (- \frac{10}{3}) = - 10 \frac{25}{3^{2}}$

Step 3.1.2.8

Raise $3$ to the power of $2$ .

$f (- \frac{10}{3}) = - 10 (\frac{25}{9})$

Step 3.1.2.9

Multiply $- 10 (\frac{25}{9})$ .

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Step 3.1.2.9.1

Combine $- 10$ and $\frac{25}{9}$ .

$f (- \frac{10}{3}) = \frac{- 10 \cdot 25}{9}$

Step 3.1.2.9.2

Multiply $- 10$ by $25$ .

$f (- \frac{10}{3}) = \frac{- 250}{9}$

Step 3.1.2.10

Move the negative in front of the fraction.

$f (- \frac{10}{3}) = - \frac{250}{9}$

Step 3.1.2.11

The final answer is $- \frac{250}{9}$ .

$- \frac{250}{9}$

Step 3.2

The point found by substituting $- \frac{10}{3}$ in $f (x) = 3 x {(x + 5)}^{2}$ is $(- \frac{10}{3}, - \frac{250}{9})$ . This point can be an inflection point.

$(- \frac{10}{3}, - \frac{250}{9})$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

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

Step 5

Substitute a value from the interval $(- \infty, - \frac{10}{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.4\overline{3}$ in the expression.

$f'' (- 3.4\overline{3}) = 18 (- 3.4\overline{3}) + 60$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $18$ by $- 3.4\overline{3}$ .

$f'' (- 3.4\overline{3}) = - 61.8 + 60$

Step 5.2.2

Add $- 61.8$ and $60$ .

$f'' (- 3.4\overline{3}) = - 1.8$

Step 5.2.3

The final answer is $- 1.8$ .

$- 1.8$

Step 5.3

At $- 3.4\overline{3}$ , the second derivative is $- 1.8$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \frac{10}{3})$

Decreasing on $(- \infty, - \frac{10}{3})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- \frac{10}{3}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 3.2\overline{3}$ in the expression.

$f'' (- 3.2\overline{3}) = 18 (- 3.2\overline{3}) + 60$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $18$ by $- 3.2\overline{3}$ .

$f'' (- 3.2\overline{3}) = - 58.2 + 60$

Step 6.2.2

Add $- 58.2$ and $60$ .

$f'' (- 3.2\overline{3}) = 1.7\overline{9}$

Step 6.2.3

The final answer is $1.7\overline{9}$ .

$1.7\overline{9}$

Step 6.3

At $- 3.2\overline{3}$ , the second derivative is $1.7\overline{9}$ . Since this is positive, the second derivative is increasing on the interval $(- \frac{10}{3}, \infty)$ .

Increasing on $(- \frac{10}{3}, \infty)$ since $f'' (x) > 0$

Step 7

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(- \frac{10}{3}, - \frac{250}{9})$ .

$(- \frac{10}{3}, - \frac{250}{9})$

Step 8