Calculus Examples

$y = \frac{1}{2} {(x - 6)}^{2}$

Step 1

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$f (x) = \frac{1}{2} \cdot ((x - 6) (x - 6))$

Step 2

Expand $(x - 6) (x - 6)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$f (x) = \frac{1}{2} \cdot (x (x - 6) - 6 (x - 6))$

Step 2.2

Apply the distributive property.

$f (x) = \frac{1}{2} \cdot (x \cdot x + x \cdot - 6 - 6 (x - 6))$

Step 2.3

Apply the distributive property.

$f (x) = \frac{1}{2} \cdot (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$f (x) = \frac{1}{2} \cdot (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6)$

Step 3.1.2

Move $- 6$ to the left of $x$ .

$f (x) = \frac{1}{2} \cdot (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6)$

Step 3.1.3

Multiply $- 6$ by $- 6$ .

$f (x) = \frac{1}{2} \cdot (x^{2} - 6 x - 6 x + 36)$

Step 3.2

Subtract $6 x$ from $- 6 x$ .

$f (x) = \frac{1}{2} \cdot (x^{2} - 12 x + 36)$

Step 4

Apply the distributive property.

$f (x) = \frac{1}{2} x^{2} + \frac{1}{2} \cdot (- 12 x) + \frac{1}{2} \cdot 36$

Step 5

Simplify.

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Step 5.1

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

$f (x) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (- 12 x) + \frac{1}{2} \cdot 36$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Factor $2$ out of $- 12 x$ .

$f (x) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (2 (- 6 x)) + \frac{1}{2} \cdot 36$

Step 5.2.2

Cancel the common factor.

$f (x) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (2 (- 6 x)) + \frac{1}{2} \cdot 36$

Step 5.2.3

Rewrite the expression.

$f (x) = \frac{x^{2}}{2} - 6 x + \frac{1}{2} \cdot 36$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Factor $2$ out of $36$ .

$f (x) = \frac{x^{2}}{2} - 6 x + \frac{1}{2} \cdot (2 (18))$

Step 5.3.2

Cancel the common factor.

$f (x) = \frac{x^{2}}{2} - 6 x + \frac{1}{2} \cdot (2 \cdot 18)$

Step 5.3.3

Rewrite the expression.

$f (x) = \frac{x^{2}}{2} - 6 x + 18$

Step 6

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 7

Find the value of $x = - \frac{b}{2 a}$ .

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Step 7.1

Substitute in the values of $a$ and $b$ .

$x = - \frac{- 6}{2 (0.5)}$

Step 7.2

Remove parentheses.

$x = - \frac{- 6}{2 (0.5)}$

Step 7.3

Simplify $- \frac{- 6}{2 (0.5)}$ .

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Step 7.3.1

Cancel the common factor of $- 6$ and $2$ .

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Step 7.3.1.1

Factor $2$ out of $- 6$ .

$x = - \frac{2 \cdot - 3}{2 \cdot 0.5}$

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Factor $2$ out of $2 \cdot 0.5$ .

$x = - \frac{2 \cdot - 3}{2 (0.5)}$

Step 7.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 3}{2 \cdot 0.5}$

Step 7.3.1.2.3

Rewrite the expression.

$x = - \frac{- 3}{0.5}$

Step 7.3.2

Simplify the expression.

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Step 7.3.2.1

Divide $- 3$ by $0.5$ .

$x = - - 6$

Step 7.3.2.2

Multiply $- 1$ by $- 6$ .

$x = 6$

Step 8

Evaluate $f (6)$ .

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Step 8.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{{(6)}^{2}}{2} - 6 \cdot 6 + 18$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $6$ to the power of $2$ .

$f (6) = \frac{36}{2} - 6 \cdot 6 + 18$

Step 8.2.1.2

Divide $36$ by $2$ .

$f (6) = 18 - 6 \cdot 6 + 18$

Step 8.2.1.3

Multiply $- 6$ by $6$ .

$f (6) = 18 - 36 + 18$

Step 8.2.2

Simplify by adding and subtracting.

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Step 8.2.2.1

Subtract $36$ from $18$ .

$f (6) = - 18 + 18$

Step 8.2.2.2

Add $- 18$ and $18$ .

$f (6) = 0$

Step 8.2.3

The final answer is $0$ .

$0$

Step 9

Use the $x$ and $y$ values to find where the minimum occurs.

$(6, 0)$

Step 10