Calculus Examples

$y = x^{5} + x^{3} - 2$

Step 1

Write $y = x^{5} + x^{3} - 2$ as a function.

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

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 $x^{5} + x^{3} - 2$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$ .

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

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

Step 2.1.3

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

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

Step 2.1.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$5 x^{4} + 3 x^{2} + 0$

Step 2.1.5

Add $5 x^{4} + 3 x^{2}$ and $0$ .

$f' (x) = 5 x^{4} + 3 x^{2}$

Step 2.2

Find the second derivative.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

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 $n = 4$ .

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

Step 2.2.2.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [3 x^{2}]$

Step 2.2.3

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

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Step 2.2.3.1

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

$20 x^{3} + 3 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$20 x^{3} + 3 (2 x)$

Step 2.2.3.3

Multiply $2$ by $3$ .

$f'' (x) = 20 x^{3} + 6 x$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $20 x^{3} + 6 x$ .

$20 x^{3} + 6 x$

Step 3

Set the second derivative equal to $0$ then solve the equation $20 x^{3} + 6 x = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$20 x^{3} + 6 x = 0$

Step 3.2

Factor $2 x$ out of $20 x^{3} + 6 x$ .

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Step 3.2.1

Factor $2 x$ out of $20 x^{3}$ .

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

Step 3.2.2

Factor $2 x$ out of $6 x$ .

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

Step 3.2.3

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

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

Step 3.3

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

$x = 0$

$10 x^{2} + 3 = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $10 x^{2} + 3$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $10 x^{2} + 3$ equal to $0$ .

$10 x^{2} + 3 = 0$

Step 3.5.2

Solve $10 x^{2} + 3 = 0$ for $x$ .

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Step 3.5.2.1

Subtract $3$ from both sides of the equation.

$10 x^{2} = - 3$

Step 3.5.2.2

Divide each term in $10 x^{2} = - 3$ by $10$ and simplify.

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Step 3.5.2.2.1

Divide each term in $10 x^{2} = - 3$ by $10$ .

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

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $10$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.2.1.2

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

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

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Move the negative in front of the fraction.

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

Step 3.5.2.3

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

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

Step 3.5.2.4

Simplify $\pm \sqrt{- \frac{3}{10}}$ .

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Step 3.5.2.4.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{3}{10}}$

Step 3.5.2.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{3}{10}}$

Step 3.5.2.4.3

Rewrite $\sqrt{\frac{3}{10}}$ as $\frac{\sqrt{3}}{\sqrt{10}}$ .

$x = \pm i \frac{\sqrt{3}}{\sqrt{10}}$

Step 3.5.2.4.4

Multiply $\frac{\sqrt{3}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$x = \pm i (\frac{\sqrt{3}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 3.5.2.4.5

Combine and simplify the denominator.

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Step 3.5.2.4.5.1

Multiply $\frac{\sqrt{3}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 3.5.2.4.5.2

Raise $\sqrt{10}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 3.5.2.4.5.3

Raise $\sqrt{10}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 3.5.2.4.5.4

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

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 3.5.2.4.5.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 3.5.2.4.5.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 3.5.2.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 3.5.2.4.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 3.5.2.4.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 3.5.2.4.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.2.4.5.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 3.5.2.4.5.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{10^{1}}$

Step 3.5.2.4.5.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{3} \sqrt{10}}{10}$

Step 3.5.2.4.6

Simplify the numerator.

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Step 3.5.2.4.6.1

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{3 \cdot 10}}{10}$

Step 3.5.2.4.6.2

Multiply $3$ by $10$ .

$x = \pm i \frac{\sqrt{30}}{10}$

Step 3.5.2.4.7

Combine $i$ and $\frac{\sqrt{30}}{10}$ .

$x = \pm \frac{i \sqrt{30}}{10}$

Step 3.5.2.5

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

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Step 3.5.2.5.1

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

$x = \frac{i \sqrt{30}}{10}$

Step 3.5.2.5.2

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

$x = - \frac{i \sqrt{30}}{10}$

Step 3.5.2.5.3

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

$x = \frac{i \sqrt{30}}{10}, - \frac{i \sqrt{30}}{10}$

Step 3.6

The final solution is all the values that make $2 x (10 x^{2} + 3) = 0$ true.

$x = 0, \frac{i \sqrt{30}}{10}, - \frac{i \sqrt{30}}{10}$

Step 4

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

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Step 4.1

Substitute $0$ in $f (x) = x^{5} + x^{3} - 2$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $0$ in the expression.

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

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

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

$f (0) = 0 + {(0)}^{3} - 2$

Step 4.1.2.1.2

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

$f (0) = 0 + 0 - 2$

Step 4.1.2.2

Simplify by adding and subtracting.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 - 2$

Step 4.1.2.2.2

Subtract $2$ from $0$ .

$f (0) = - 2$

Step 4.1.2.3

The final answer is $- 2$ .

$- 2$

Step 4.2

The point found by substituting $0$ in $f (x) = x^{5} + x^{3} - 2$ is $(0, - 2)$ . This point can be an inflection point.

$(0, - 2)$

Step 5

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

$(- \infty, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.1$ in the expression.

$f'' (- 0.1) = 20 {(- 0.1)}^{3} + 6 (- 0.1)$

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 $- 0.1$ to the power of $3$ .

$f'' (- 0.1) = 20 \cdot - 0.001 + 6 (- 0.1)$

Step 6.2.1.2

Multiply $20$ by $- 0.001$ .

$f'' (- 0.1) = - 0.02 + 6 (- 0.1)$

Step 6.2.1.3

Multiply $6$ by $- 0.1$ .

$f'' (- 0.1) = - 0.02 - 0.6$

Step 6.2.2

Subtract $0.6$ from $- 0.02$ .

$f'' (- 0.1) = - 0.62$

Step 6.2.3

The final answer is $- 0.62$ .

$- 0.62$

Step 6.3

At $- 0.1$ , the second derivative is $- 0.62$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, 0)$

Decreasing on $(- \infty, 0)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = 20 {(0.1)}^{3} + 6 (0.1)$

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 $0.1$ to the power of $3$ .

$f'' (0.1) = 20 \cdot 0.001 + 6 (0.1)$

Step 7.2.1.2

Multiply $20$ by $0.001$ .

$f'' (0.1) = 0.02 + 6 (0.1)$

Step 7.2.1.3

Multiply $6$ by $0.1$ .

$f'' (0.1) = 0.02 + 0.6$

Step 7.2.2

Add $0.02$ and $0.6$ .

$f'' (0.1) = 0.62$

Step 7.2.3

The final answer is $0.62$ .

$0.62$

Step 7.3

At $0.1$ , the second derivative is $0.62$ . Since this is positive, the second derivative is increasing on the interval $(0, \infty)$ .

Increasing on $(0, \infty)$ 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. The inflection point in this case is $(0, - 2)$ .

$(0, - 2)$

Step 9