Calculus Examples

$3 x^{4} - 96 x + 7$

Step 1

Write $3 x^{4} - 96 x + 7$ as a function.

$f (x) = 3 x^{4} - 96 x + 7$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{4} - 96 x + 7$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$

Step 2.2

Evaluate $\frac{d}{d x} [3 x^{4}]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$

Step 2.2.2

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

$3 (4 x^{3}) + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$

Step 2.2.3

Multiply $4$ by $3$ .

$12 x^{3} + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$

Step 2.3

Evaluate $\frac{d}{d x} [- 96 x]$ .

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Step 2.3.1

Since $- 96$ is constant with respect to $x$ , the derivative of $- 96 x$ with respect to $x$ is $- 96 \frac{d}{d x} [x]$ .

$12 x^{3} - 96 \frac{d}{d x} [x] + \frac{d}{d x} [7]$

Step 2.3.2

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

$12 x^{3} - 96 \cdot 1 + \frac{d}{d x} [7]$

Step 2.3.3

Multiply $- 96$ by $1$ .

$12 x^{3} - 96 + \frac{d}{d x} [7]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$12 x^{3} - 96 + 0$

Step 2.4.2

Add $12 x^{3} - 96$ and $0$ .

$12 x^{3} - 96$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $12 x^{3} - 96$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [- 96]$ .

$f'' (x) = \frac{d}{d x} (12 x^{3}) + \frac{d}{d x} (- 96)$

Step 3.2

Evaluate $\frac{d}{d x} [12 x^{3}]$ .

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Step 3.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3}$ with respect to $x$ is $12 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 12 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 96)$

Step 3.2.2

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

$f'' (x) = 12 (3 x^{2}) + \frac{d}{d x} (- 96)$

Step 3.2.3

Multiply $3$ by $12$ .

$f'' (x) = 36 x^{2} + \frac{d}{d x} (- 96)$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $- 96$ is constant with respect to $x$ , the derivative of $- 96$ with respect to $x$ is $0$ .

$f'' (x) = 36 x^{2} + 0$

Step 3.3.2

Add $36 x^{2}$ and $0$ .

$f'' (x) = 36 x^{2}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$12 x^{3} - 96 = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $3 x^{4} - 96 x + 7$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$

Step 5.1.2

Evaluate $\frac{d}{d x} [3 x^{4}]$ .

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Step 5.1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$

Step 5.1.2.2

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

$3 (4 x^{3}) + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$

Step 5.1.2.3

Multiply $4$ by $3$ .

$12 x^{3} + \frac{d}{d x} [- 96 x] + \frac{d}{d x} [7]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- 96 x]$ .

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Step 5.1.3.1

Since $- 96$ is constant with respect to $x$ , the derivative of $- 96 x$ with respect to $x$ is $- 96 \frac{d}{d x} [x]$ .

$12 x^{3} - 96 \frac{d}{d x} [x] + \frac{d}{d x} [7]$

Step 5.1.3.2

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

$12 x^{3} - 96 \cdot 1 + \frac{d}{d x} [7]$

Step 5.1.3.3

Multiply $- 96$ by $1$ .

$12 x^{3} - 96 + \frac{d}{d x} [7]$

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$12 x^{3} - 96 + 0$

Step 5.1.4.2

Add $12 x^{3} - 96$ and $0$ .

$f' (x) = 12 x^{3} - 96$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{3} - 96$ .

$12 x^{3} - 96$

Step 6

Set the first derivative equal to $0$ then solve the equation $12 x^{3} - 96 = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$12 x^{3} - 96 = 0$

Step 6.2

Add $96$ to both sides of the equation.

$12 x^{3} = 96$

Step 6.3

Subtract $96$ from both sides of the equation.

$12 x^{3} - 96 = 0$

Step 6.4

Factor the left side of the equation.

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Step 6.4.1

Factor $12$ out of $12 x^{3} - 96$ .

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Step 6.4.1.1

Factor $12$ out of $12 x^{3}$ .

$12 (x^{3}) - 96 = 0$

Step 6.4.1.2

Factor $12$ out of $- 96$ .

$12 x^{3} + 12 \cdot - 8 = 0$

Step 6.4.1.3

Factor $12$ out of $12 x^{3} + 12 \cdot - 8$ .

$12 (x^{3} - 8) = 0$

Step 6.4.2

Rewrite $8$ as $2^{3}$ .

$12 (x^{3} - 2^{3}) = 0$

Step 6.4.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 2$ .

$12 ((x - 2) (x^{2} + x \cdot 2 + 2^{2})) = 0$

Step 6.4.4

Factor.

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Step 6.4.4.1

Simplify.

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Step 6.4.4.1.1

Move $2$ to the left of $x$ .

$12 ((x - 2) (x^{2} + 2 x + 2^{2})) = 0$

Step 6.4.4.1.2

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

$12 ((x - 2) (x^{2} + 2 x + 4)) = 0$

Step 6.4.4.2

Remove unnecessary parentheses.

$12 (x - 2) (x^{2} + 2 x + 4) = 0$

Step 6.5

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

$x - 2 = 0$

$x^{2} + 2 x + 4 = 0$

Step 6.6

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 6.6.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 6.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6.7

Set $x^{2} + 2 x + 4$ equal to $0$ and solve for $x$ .

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Step 6.7.1

Set $x^{2} + 2 x + 4$ equal to $0$ .

$x^{2} + 2 x + 4 = 0$

Step 6.7.2

Solve $x^{2} + 2 x + 4 = 0$ for $x$ .

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Step 6.7.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.7.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 6.7.2.3

Simplify.

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Step 6.7.2.3.1

Simplify the numerator.

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Step 6.7.2.3.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 6.7.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 6.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 6.7.2.3.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 6.7.2.3.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 6.7.2.3.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 6.7.2.3.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 6.7.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 6.7.2.3.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.7.2.3.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 6.7.2.3.1.7.2

Rewrite $4$ as $2^{2}$ .

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

Step 6.7.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 6.7.2.3.1.9

Move $2$ to the left of $i$ .

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

Step 6.7.2.3.2

Multiply $2$ by $1$ .

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

Step 6.7.2.3.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 6.7.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.7.2.4.1

Simplify the numerator.

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Step 6.7.2.4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 6.7.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 6.7.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 6.7.2.4.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 6.7.2.4.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 6.7.2.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 6.7.2.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 6.7.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 6.7.2.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.7.2.4.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 6.7.2.4.1.7.2

Rewrite $4$ as $2^{2}$ .

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

Step 6.7.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 6.7.2.4.1.9

Move $2$ to the left of $i$ .

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

Step 6.7.2.4.2

Multiply $2$ by $1$ .

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

Step 6.7.2.4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 6.7.2.4.4

Change the $\pm$ to $+$ .

$x = - 1 + i \sqrt{3}$

Step 6.7.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.7.2.5.1

Simplify the numerator.

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Step 6.7.2.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 6.7.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 6.7.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 6.7.2.5.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 6.7.2.5.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 6.7.2.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 6.7.2.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 6.7.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 6.7.2.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.7.2.5.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 6.7.2.5.1.7.2

Rewrite $4$ as $2^{2}$ .

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

Step 6.7.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 6.7.2.5.1.9

Move $2$ to the left of $i$ .

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

Step 6.7.2.5.2

Multiply $2$ by $1$ .

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

Step 6.7.2.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 6.7.2.5.4

Change the $\pm$ to $-$ .

$x = - 1 - i \sqrt{3}$

Step 6.7.2.6

The final answer is the combination of both solutions.

$x = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 6.8

The final solution is all the values that make $12 (x - 2) (x^{2} + 2 x + 4) = 0$ true.

$x = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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.

Step 8

Critical points to evaluate.

$x = 2$

Step 9

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$36 {(2)}^{2}$

Step 10

Evaluate the second derivative.

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Step 10.1

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

$36 \cdot 4$

Step 10.2

Multiply $36$ by $4$ .

$144$

Step 11

$x = 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 2$ is a local minimum

Step 12

Find the y-value when $x = 2$ .

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Step 12.1

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

$f (2) = 3 {(2)}^{4} - 96 \cdot 2 + 7$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Raise $2$ to the power of $4$ .

$f (2) = 3 \cdot 16 - 96 \cdot 2 + 7$

Step 12.2.1.2

Multiply $3$ by $16$ .

$f (2) = 48 - 96 \cdot 2 + 7$

Step 12.2.1.3

Multiply $- 96$ by $2$ .

$f (2) = 48 - 192 + 7$

Step 12.2.2

Simplify by adding and subtracting.

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Step 12.2.2.1

Subtract $192$ from $48$ .

$f (2) = - 144 + 7$

Step 12.2.2.2

Add $- 144$ and $7$ .

$f (2) = - 137$

Step 12.2.3

The final answer is $- 137$ .

$y = - 137$

Step 13

These are the local extrema for $f (x) = 3 x^{4} - 96 x + 7$ .

$(2, - 137)$ is a local minima

Step 14