Calculus Examples

$f (x) = 12 x^{2} - 2 x^{3} + 3 y^{2} + 6 x y$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $12 x^{2} - 2 x^{3} + 3 y^{2} + 6 x y$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$ .

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 1.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 1.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 1.2.2

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

$12 (2 x) + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 1.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 1.3

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

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Step 1.3.1

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

$24 x - 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 1.3.2

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

$24 x - 2 (3 x^{2}) + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 1.3.3

Multiply $3$ by $- 2$ .

$24 x - 6 x^{2} + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 1.4

Since $3 y^{2}$ is constant with respect to $x$ , the derivative of $3 y^{2}$ with respect to $x$ is $0$ .

$24 x - 6 x^{2} + 0 + \frac{d}{d x} [6 x y]$

Step 1.5

Evaluate $\frac{d}{d x} [6 x y]$ .

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Step 1.5.1

Since $6 y$ is constant with respect to $x$ , the derivative of $6 x y$ with respect to $x$ is $6 y \frac{d}{d x} [x]$ .

$24 x - 6 x^{2} + 0 + 6 y \frac{d}{d x} [x]$

Step 1.5.2

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

$24 x - 6 x^{2} + 0 + 6 y \cdot 1$

Step 1.5.3

Multiply $6$ by $1$ .

$24 x - 6 x^{2} + 0 + 6 y$

Step 1.6

Simplify.

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Step 1.6.1

Add $24 x - 6 x^{2}$ and $0$ .

$24 x - 6 x^{2} + 6 y$

Step 1.6.2

Reorder terms.

$- 6 x^{2} + 24 x + 6 y$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 6 x^{2} + 24 x + 6 y$ with respect to $x$ is $\frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [6 y]$ .

$f'' (x) = \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (24 x) + \frac{d}{d x} (6 y)$

Step 2.2

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

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Step 2.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - 6 \frac{d}{d x} x^{2} + \frac{d}{d x} (24 x) + \frac{d}{d x} (6 y)$

Step 2.2.2

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

$f'' (x) = - 6 (2 x) + \frac{d}{d x} (24 x) + \frac{d}{d x} (6 y)$

Step 2.2.3

Multiply $2$ by $- 6$ .

$f'' (x) = - 12 x + \frac{d}{d x} (24 x) + \frac{d}{d x} (6 y)$

Step 2.3

Evaluate $\frac{d}{d x} [24 x]$ .

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Step 2.3.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$f'' (x) = - 12 x + 24 \frac{d}{d x} (x) + \frac{d}{d x} (6 y)$

Step 2.3.2

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

$f'' (x) = - 12 x + 24 \cdot 1 + \frac{d}{d x} (6 y)$

Step 2.3.3

Multiply $24$ by $1$ .

$f'' (x) = - 12 x + 24 + \frac{d}{d x} (6 y)$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $6 y$ is constant with respect to $x$ , the derivative of $6 y$ with respect to $x$ is $0$ .

$f'' (x) = - 12 x + 24 + 0$

Step 2.4.2

Add $- 12 x + 24$ and $0$ .

$f'' (x) = - 12 x + 24$

Step 3

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

$- 6 x^{2} + 24 x + 6 y = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 4.1.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 4.1.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 4.1.2.2

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

$12 (2 x) + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 4.1.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 4.1.3

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

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Step 4.1.3.1

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

$24 x - 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 4.1.3.2

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

$24 x - 2 (3 x^{2}) + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 4.1.3.3

Multiply $3$ by $- 2$ .

$24 x - 6 x^{2} + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 x y]$

Step 4.1.4

Since $3 y^{2}$ is constant with respect to $x$ , the derivative of $3 y^{2}$ with respect to $x$ is $0$ .

$24 x - 6 x^{2} + 0 + \frac{d}{d x} [6 x y]$

Step 4.1.5

Evaluate $\frac{d}{d x} [6 x y]$ .

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Step 4.1.5.1

Since $6 y$ is constant with respect to $x$ , the derivative of $6 x y$ with respect to $x$ is $6 y \frac{d}{d x} [x]$ .

$24 x - 6 x^{2} + 0 + 6 y \frac{d}{d x} [x]$

Step 4.1.5.2

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

$24 x - 6 x^{2} + 0 + 6 y \cdot 1$

Step 4.1.5.3

Multiply $6$ by $1$ .

$24 x - 6 x^{2} + 0 + 6 y$

Step 4.1.6

Simplify.

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Step 4.1.6.1

Add $24 x - 6 x^{2}$ and $0$ .

$24 x - 6 x^{2} + 6 y$

Step 4.1.6.2

Reorder terms.

$f' (x) = - 6 x^{2} + 24 x + 6 y$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 6 x^{2} + 24 x + 6 y$ .

$- 6 x^{2} + 24 x + 6 y$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 6 x^{2} + 24 x + 6 y = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- 6 x^{2} + 24 x + 6 y = 0$

Step 5.2

Use the quadratic formula to find the solutions.

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

Step 5.3

Substitute the values $a = - 6$ , $b = 24$ , and $c = 6 y$ into the quadratic formula and solve for $x$ .

$\frac{- 24 \pm \sqrt{24^{2} - 4 \cdot (- 6 \cdot (6 y))}}{2 \cdot - 6}$

Step 5.4

Simplify.

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Step 5.4.1

Simplify the numerator.

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Step 5.4.1.1

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot - 6 \cdot (6 y)}}{2 \cdot - 6}$

Step 5.4.1.2

Multiply $- 4 \cdot - 6 \cdot 6$ .

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Step 5.4.1.2.1

Multiply $- 4$ by $- 6$ .

$x = \frac{- 24 \pm \sqrt{576 + 24 \cdot (6 y)}}{2 \cdot - 6}$

Step 5.4.1.2.2

Multiply $24$ by $6$ .

$x = \frac{- 24 \pm \sqrt{576 + 144 y}}{2 \cdot - 6}$

Step 5.4.1.3

Factor $144$ out of $576 + 144 y$ .

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Step 5.4.1.3.1

Factor $144$ out of $576$ .

$x = \frac{- 24 \pm \sqrt{144 \cdot 4 + 144 y}}{2 \cdot - 6}$

Step 5.4.1.3.2

Factor $144$ out of $144 \cdot 4 + 144 y$ .

$x = \frac{- 24 \pm \sqrt{144 (4 + y)}}{2 \cdot - 6}$

Step 5.4.1.4

Rewrite $144 (4 + y)$ as $12^{2} (2^{2} + y)$ .

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Step 5.4.1.4.1

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 24 \pm \sqrt{12^{2} (4 + y)}}{2 \cdot - 6}$

Step 5.4.1.4.2

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

$x = \frac{- 24 \pm \sqrt{12^{2} (2^{2} + y)}}{2 \cdot - 6}$

Step 5.4.1.5

Pull terms out from under the radical.

$x = \frac{- 24 \pm 12 \sqrt{2^{2} + y}}{2 \cdot - 6}$

Step 5.4.1.6

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

$x = \frac{- 24 \pm 12 \sqrt{4 + y}}{2 \cdot - 6}$

Step 5.4.2

Multiply $2$ by $- 6$ .

$x = \frac{- 24 \pm 12 \sqrt{4 + y}}{- 12}$

Step 5.4.3

Simplify $\frac{- 24 \pm 12 \sqrt{4 + y}}{- 12}$ .

$x = 2 \pm \sqrt{4 + y}$

Step 5.5

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

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot - 6 \cdot (6 y)}}{2 \cdot - 6}$

Step 5.5.1.2

Multiply $- 4 \cdot - 6 \cdot 6$ .

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Step 5.5.1.2.1

Multiply $- 4$ by $- 6$ .

$x = \frac{- 24 \pm \sqrt{576 + 24 \cdot (6 y)}}{2 \cdot - 6}$

Step 5.5.1.2.2

Multiply $24$ by $6$ .

$x = \frac{- 24 \pm \sqrt{576 + 144 y}}{2 \cdot - 6}$

Step 5.5.1.3

Factor $144$ out of $576 + 144 y$ .

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Step 5.5.1.3.1

Factor $144$ out of $576$ .

$x = \frac{- 24 \pm \sqrt{144 \cdot 4 + 144 y}}{2 \cdot - 6}$

Step 5.5.1.3.2

Factor $144$ out of $144 \cdot 4 + 144 y$ .

$x = \frac{- 24 \pm \sqrt{144 (4 + y)}}{2 \cdot - 6}$

Step 5.5.1.4

Rewrite $144 (4 + y)$ as $12^{2} (2^{2} + y)$ .

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Step 5.5.1.4.1

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 24 \pm \sqrt{12^{2} (4 + y)}}{2 \cdot - 6}$

Step 5.5.1.4.2

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

$x = \frac{- 24 \pm \sqrt{12^{2} (2^{2} + y)}}{2 \cdot - 6}$

Step 5.5.1.5

Pull terms out from under the radical.

$x = \frac{- 24 \pm 12 \sqrt{2^{2} + y}}{2 \cdot - 6}$

Step 5.5.1.6

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

$x = \frac{- 24 \pm 12 \sqrt{4 + y}}{2 \cdot - 6}$

Step 5.5.2

Multiply $2$ by $- 6$ .

$x = \frac{- 24 \pm 12 \sqrt{4 + y}}{- 12}$

Step 5.5.3

Simplify $\frac{- 24 \pm 12 \sqrt{4 + y}}{- 12}$ .

$x = 2 \pm \sqrt{4 + y}$

Step 5.5.4

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{4 + y}$

Step 5.6

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

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot - 6 \cdot (6 y)}}{2 \cdot - 6}$

Step 5.6.1.2

Multiply $- 4 \cdot - 6 \cdot 6$ .

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Step 5.6.1.2.1

Multiply $- 4$ by $- 6$ .

$x = \frac{- 24 \pm \sqrt{576 + 24 \cdot (6 y)}}{2 \cdot - 6}$

Step 5.6.1.2.2

Multiply $24$ by $6$ .

$x = \frac{- 24 \pm \sqrt{576 + 144 y}}{2 \cdot - 6}$

Step 5.6.1.3

Factor $144$ out of $576 + 144 y$ .

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Step 5.6.1.3.1

Factor $144$ out of $576$ .

$x = \frac{- 24 \pm \sqrt{144 \cdot 4 + 144 y}}{2 \cdot - 6}$

Step 5.6.1.3.2

Factor $144$ out of $144 \cdot 4 + 144 y$ .

$x = \frac{- 24 \pm \sqrt{144 (4 + y)}}{2 \cdot - 6}$

Step 5.6.1.4

Rewrite $144 (4 + y)$ as $12^{2} (2^{2} + y)$ .

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Step 5.6.1.4.1

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 24 \pm \sqrt{12^{2} (4 + y)}}{2 \cdot - 6}$

Step 5.6.1.4.2

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

$x = \frac{- 24 \pm \sqrt{12^{2} (2^{2} + y)}}{2 \cdot - 6}$

Step 5.6.1.5

Pull terms out from under the radical.

$x = \frac{- 24 \pm 12 \sqrt{2^{2} + y}}{2 \cdot - 6}$

Step 5.6.1.6

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

$x = \frac{- 24 \pm 12 \sqrt{4 + y}}{2 \cdot - 6}$

Step 5.6.2

Multiply $2$ by $- 6$ .

$x = \frac{- 24 \pm 12 \sqrt{4 + y}}{- 12}$

Step 5.6.3

Simplify $\frac{- 24 \pm 12 \sqrt{4 + y}}{- 12}$ .

$x = 2 \pm \sqrt{4 + y}$

Step 5.6.4

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{4 + y}$

Step 5.7

The final answer is the combination of both solutions.

$x = 2 + \sqrt{4 + y}$

$x = 2 - \sqrt{4 + y}$

$x = 2 + \sqrt{4 + y}$

$x = 2 - \sqrt{4 + y}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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 7

Critical points to evaluate.

$x = 2 + \sqrt{4 + y}$

$x = 2 - \sqrt{4 + y}$

Step 8

Evaluate the second derivative at $x = 2 + \sqrt{4 + y}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 12 (2 + \sqrt{4 + y}) + 24$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Apply the distributive property.

$- 12 \cdot 2 - 12 \sqrt{4 + y} + 24$

Step 9.1.2

Multiply $- 12$ by $2$ .

$- 24 - 12 \sqrt{4 + y} + 24$

Step 9.2

Combine the opposite terms in $- 24 - 12 \sqrt{4 + y} + 24$ .

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Step 9.2.1

Add $- 24$ and $24$ .

$0 - 12 \sqrt{4 + y}$

Step 9.2.2

Subtract $12 \sqrt{4 + y}$ from $0$ .

$- 12 \sqrt{4 + y}$

Step 10

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 11