Calculus Examples

$\frac{1}{4} x^{4} + 5 x^{3} + \frac{75}{2} x^{2}$

Step 1

Write $\frac{1}{4} x^{4} + 5 x^{3} + \frac{75}{2} x^{2}$ as a function.

$f (x) = \frac{1}{4} x^{4} + 5 x^{3} + \frac{75}{2} x^{2}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $\frac{1}{4} x^{4} + 5 x^{3} + \frac{75}{2} x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [\frac{75}{2} x^{2}]$ .

$\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.2

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

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Step 2.1.2.1

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

$\frac{1}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{4} (4 x^{3}) + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.2.3

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

$\frac{4}{4} x^{3} + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.2.4

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

$\frac{4 x^{3}}{4} + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.2.5

Cancel the common factor of $4$ .

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Step 2.1.2.5.1

Cancel the common factor.

$\frac{4 x^{3}}{4} + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.2.5.2

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

$x^{3} + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [5 x^{3}]$ .

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Step 2.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$x^{3} + 5 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.3.2

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

$x^{3} + 5 (3 x^{2}) + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.3.3

Multiply $3$ by $5$ .

$x^{3} + 15 x^{2} + \frac{d}{d x} [\frac{75}{2} x^{2}]$

Step 2.1.4

Evaluate $\frac{d}{d x} [\frac{75}{2} x^{2}]$ .

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Step 2.1.4.1

Since $\frac{75}{2}$ is constant with respect to $x$ , the derivative of $\frac{75}{2} x^{2}$ with respect to $x$ is $\frac{75}{2} \frac{d}{d x} [x^{2}]$ .

$x^{3} + 15 x^{2} + \frac{75}{2} \frac{d}{d x} [x^{2}]$

Step 2.1.4.2

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

$x^{3} + 15 x^{2} + \frac{75}{2} (2 x)$

Step 2.1.4.3

Combine $2$ and $\frac{75}{2}$ .

$x^{3} + 15 x^{2} + \frac{2 \cdot 75}{2} x$

Step 2.1.4.4

Multiply $2$ by $75$ .

$x^{3} + 15 x^{2} + \frac{150}{2} x$

Step 2.1.4.5

Combine $\frac{150}{2}$ and $x$ .

$x^{3} + 15 x^{2} + \frac{150 x}{2}$

Step 2.1.4.6

Cancel the common factor of $150$ and $2$ .

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Step 2.1.4.6.1

Factor $2$ out of $150 x$ .

$x^{3} + 15 x^{2} + \frac{2 (75 x)}{2}$

Step 2.1.4.6.2

Cancel the common factors.

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Step 2.1.4.6.2.1

Factor $2$ out of $2$ .

$x^{3} + 15 x^{2} + \frac{2 (75 x)}{2 (1)}$

Step 2.1.4.6.2.2

Cancel the common factor.

$x^{3} + 15 x^{2} + \frac{2 (75 x)}{2 \cdot 1}$

Step 2.1.4.6.2.3

Rewrite the expression.

$x^{3} + 15 x^{2} + \frac{75 x}{1}$

Step 2.1.4.6.2.4

Divide $75 x$ by $1$ .

$f' (x) = x^{3} + 15 x^{2} + 75 x$

Step 2.2

Find the second derivative.

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Step 2.2.1

Differentiate.

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Step 2.2.1.1

By the Sum Rule, the derivative of $x^{3} + 15 x^{2} + 75 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [75 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [75 x]$

Step 2.2.1.2

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

$3 x^{2} + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [75 x]$

Step 2.2.2

Evaluate $\frac{d}{d x} [15 x^{2}]$ .

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Step 2.2.2.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} + 15 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [75 x]$

Step 2.2.2.2

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

$3 x^{2} + 15 (2 x) + \frac{d}{d x} [75 x]$

Step 2.2.2.3

Multiply $2$ by $15$ .

$3 x^{2} + 30 x + \frac{d}{d x} [75 x]$

Step 2.2.3

Evaluate $\frac{d}{d x} [75 x]$ .

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Step 2.2.3.1

Since $75$ is constant with respect to $x$ , the derivative of $75 x$ with respect to $x$ is $75 \frac{d}{d x} [x]$ .

$3 x^{2} + 30 x + 75 \frac{d}{d x} [x]$

Step 2.2.3.2

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

$3 x^{2} + 30 x + 75 \cdot 1$

Step 2.2.3.3

Multiply $75$ by $1$ .

$f'' (x) = 3 x^{2} + 30 x + 75$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 30 x + 75$ .

$3 x^{2} + 30 x + 75$

Step 3

Set the second derivative equal to $0$ then solve the equation $3 x^{2} + 30 x + 75 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$3 x^{2} + 30 x + 75 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $3$ out of $3 x^{2} + 30 x + 75$ .

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Step 3.2.1.1

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

$3 (x^{2}) + 30 x + 75 = 0$

Step 3.2.1.2

Factor $3$ out of $30 x$ .

$3 (x^{2}) + 3 (10 x) + 75 = 0$

Step 3.2.1.3

Factor $3$ out of $75$ .

$3 x^{2} + 3 (10 x) + 3 \cdot 25 = 0$

Step 3.2.1.4

Factor $3$ out of $3 x^{2} + 3 (10 x)$ .

$3 (x^{2} + 10 x) + 3 \cdot 25 = 0$

Step 3.2.1.5

Factor $3$ out of $3 (x^{2} + 10 x) + 3 \cdot 25$ .

$3 (x^{2} + 10 x + 25) = 0$

Step 3.2.2

Factor using the perfect square rule.

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Step 3.2.2.1

Rewrite $25$ as $5^{2}$ .

$3 (x^{2} + 10 x + 5^{2}) = 0$

Step 3.2.2.2

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

$10 x = 2 \cdot x \cdot 5$

Step 3.2.2.3

Rewrite the polynomial.

$3 (x^{2} + 2 \cdot x \cdot 5 + 5^{2}) = 0$

Step 3.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$3 {(x + 5)}^{2} = 0$

Step 3.3

Divide each term in $3 {(x + 5)}^{2} = 0$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 {(x + 5)}^{2} = 0$ by $3$ .

$\frac{3 {(x + 5)}^{2}}{3} = \frac{0}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 {(x + 5)}^{2}}{3} = \frac{0}{3}$

Step 3.3.2.1.2

Divide ${(x + 5)}^{2}$ by $1$ .

${(x + 5)}^{2} = \frac{0}{3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $3$ .

${(x + 5)}^{2} = 0$

Step 3.4

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 3.5

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4

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

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Step 4.1

Substitute $- 5$ in $f (x) = \frac{1}{4} x^{4} + 5 x^{3} + \frac{75}{2} x^{2}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = \frac{1}{4} \cdot {(- 5)}^{4} + 5 {(- 5)}^{3} + \frac{75}{2} \cdot {(- 5)}^{2}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raise $- 5$ to the power of $4$ .

$f (- 5) = \frac{1}{4} \cdot 625 + 5 {(- 5)}^{3} + \frac{75}{2} \cdot {(- 5)}^{2}$

Step 4.1.2.1.2

Combine $\frac{1}{4}$ and $625$ .

$f (- 5) = \frac{625}{4} + 5 {(- 5)}^{3} + \frac{75}{2} \cdot {(- 5)}^{2}$

Step 4.1.2.1.3

Raise $- 5$ to the power of $3$ .

$f (- 5) = \frac{625}{4} + 5 \cdot - 125 + \frac{75}{2} \cdot {(- 5)}^{2}$

Step 4.1.2.1.4

Multiply $5$ by $- 125$ .

$f (- 5) = \frac{625}{4} - 625 + \frac{75}{2} \cdot {(- 5)}^{2}$

Step 4.1.2.1.5

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

$f (- 5) = \frac{625}{4} - 625 + \frac{75}{2} \cdot 25$

Step 4.1.2.1.6

Multiply $\frac{75}{2} \cdot 25$ .

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Step 4.1.2.1.6.1

Combine $\frac{75}{2}$ and $25$ .

$f (- 5) = \frac{625}{4} - 625 + \frac{75 \cdot 25}{2}$

Step 4.1.2.1.6.2

Multiply $75$ by $25$ .

$f (- 5) = \frac{625}{4} - 625 + \frac{1875}{2}$

Step 4.1.2.2

Find the common denominator.

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Step 4.1.2.2.1

Write $- 625$ as a fraction with denominator $1$ .

$f (- 5) = \frac{625}{4} + \frac{- 625}{1} + \frac{1875}{2}$

Step 4.1.2.2.2

Multiply $\frac{- 625}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{625}{4} + \frac{- 625}{1} \cdot \frac{4}{4} + \frac{1875}{2}$

Step 4.1.2.2.3

Multiply $\frac{- 625}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{625}{4} + \frac{- 625 \cdot 4}{4} + \frac{1875}{2}$

Step 4.1.2.2.4

Multiply $\frac{1875}{2}$ by $\frac{2}{2}$ .

$f (- 5) = \frac{625}{4} + \frac{- 625 \cdot 4}{4} + \frac{1875}{2} \cdot \frac{2}{2}$

Step 4.1.2.2.5

Multiply $\frac{1875}{2}$ by $\frac{2}{2}$ .

$f (- 5) = \frac{625}{4} + \frac{- 625 \cdot 4}{4} + \frac{1875 \cdot 2}{2 \cdot 2}$

Step 4.1.2.2.6

Multiply $2$ by $2$ .

$f (- 5) = \frac{625}{4} + \frac{- 625 \cdot 4}{4} + \frac{1875 \cdot 2}{4}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$f (- 5) = \frac{625 - 625 \cdot 4 + 1875 \cdot 2}{4}$

Step 4.1.2.4

Simplify each term.

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Step 4.1.2.4.1

Multiply $- 625$ by $4$ .

$f (- 5) = \frac{625 - 2500 + 1875 \cdot 2}{4}$

Step 4.1.2.4.2

Multiply $1875$ by $2$ .

$f (- 5) = \frac{625 - 2500 + 3750}{4}$

Step 4.1.2.5

Simplify by adding and subtracting.

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Step 4.1.2.5.1

Subtract $2500$ from $625$ .

$f (- 5) = \frac{- 1875 + 3750}{4}$

Step 4.1.2.5.2

Add $- 1875$ and $3750$ .

$f (- 5) = \frac{1875}{4}$

Step 4.1.2.6

The final answer is $\frac{1875}{4}$ .

$\frac{1875}{4}$

Step 4.2

The point found by substituting $- 5$ in $f (x) = \frac{1}{4} x^{4} + 5 x^{3} + \frac{75}{2} x^{2}$ is $(- 5, \frac{1875}{4})$ . This point can be an inflection point.

$(- 5, \frac{1875}{4})$

Step 5

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

$(- \infty, - 5) \cup (- 5, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 5)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 5.1$ in the expression.

$f'' (- 5.1) = 3 {(- 5.1)}^{2} + 30 (- 5.1) + 75$

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 $- 5.1$ to the power of $2$ .

$f'' (- 5.1) = 3 \cdot 26.01 + 30 (- 5.1) + 75$

Step 6.2.1.2

Multiply $3$ by $26.01$ .

$f'' (- 5.1) = 78.03 + 30 (- 5.1) + 75$

Step 6.2.1.3

Multiply $30$ by $- 5.1$ .

$f'' (- 5.1) = 78.03 - 153 + 75$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $153$ from $78.03$ .

$f'' (- 5.1) = - 74.97 + 75$

Step 6.2.2.2

Add $- 74.97$ and $75$ .

$f'' (- 5.1) = 0.03$

Step 6.2.3

The final answer is $0.03$ .

$0.03$

Step 6.3

At $- 5.1$ , the second derivative is $0.03$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 5)$ .

Increasing on $(- \infty, - 5)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(- 5, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 4.9$ in the expression.

$f'' (- 4.9) = 3 {(- 4.9)}^{2} + 30 (- 4.9) + 75$

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 $- 4.9$ to the power of $2$ .

$f'' (- 4.9) = 3 \cdot 24.01 + 30 (- 4.9) + 75$

Step 7.2.1.2

Multiply $3$ by $24.01$ .

$f'' (- 4.9) = 72.03 + 30 (- 4.9) + 75$

Step 7.2.1.3

Multiply $30$ by $- 4.9$ .

$f'' (- 4.9) = 72.03 - 147 + 75$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $147$ from $72.03$ .

$f'' (- 4.9) = - 74.97 + 75$

Step 7.2.2.2

Add $- 74.97$ and $75$ .

$f'' (- 4.9) = 0.03$

Step 7.2.3

The final answer is $0.03$ .

$0.03$

Step 7.3

At $- 4.9$ , the second derivative is $0.03$ . Since this is positive, the second derivative is increasing on the interval $(- 5, \infty)$ .

Increasing on $(- 5, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. There are no points on the graph that satisfy these requirements.

No Inflection Points