Calculus Examples

$f (x) = 3 (x - 9) (x - 6)$

Step 1

Simplify by multiplying through.

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Step 1.1

Apply the distributive property.

$f (x) = (3 x + 3 \cdot - 9) (x - 6)$

Step 1.2

Multiply $3$ by $- 9$ .

$f (x) = (3 x - 27) (x - 6)$

Step 2

Expand $(3 x - 27) (x - 6)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$f (x) = 3 x (x - 6) - 27 (x - 6)$

Step 2.2

Apply the distributive property.

$f (x) = 3 x \cdot x + 3 x \cdot - 6 - 27 (x - 6)$

Step 2.3

Apply the distributive property.

$f (x) = 3 x \cdot x + 3 x \cdot - 6 - 27 x - 27 \cdot - 6$

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 $x$ by adding the exponents.

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Step 3.1.1.1

Move $x$ .

$f (x) = 3 (x \cdot x) + 3 x \cdot - 6 - 27 x - 27 \cdot - 6$

Step 3.1.1.2

Multiply $x$ by $x$ .

$f (x) = 3 x^{2} + 3 x \cdot - 6 - 27 x - 27 \cdot - 6$

Step 3.1.2

Multiply $- 6$ by $3$ .

$f (x) = 3 x^{2} - 18 x - 27 x - 27 \cdot - 6$

Step 3.1.3

Multiply $- 27$ by $- 6$ .

$f (x) = 3 x^{2} - 18 x - 27 x + 162$

Step 3.2

Subtract $27 x$ from $- 18 x$ .

$f (x) = 3 x^{2} - 45 x + 162$

Step 4

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 5

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

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Step 5.1

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

$x = - \frac{- 45}{2 (3)}$

Step 5.2

Remove parentheses.

$x = - \frac{- 45}{2 (3)}$

Step 5.3

Simplify $- \frac{- 45}{2 (3)}$ .

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Step 5.3.1

Cancel the common factor of $- 45$ and $3$ .

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Step 5.3.1.1

Factor $3$ out of $- 45$ .

$x = - \frac{3 \cdot - 15}{2 \cdot 3}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $3$ out of $2 \cdot 3$ .

$x = - \frac{3 \cdot - 15}{3 \cdot 2}$

Step 5.3.1.2.2

Cancel the common factor.

$x = - \frac{3 \cdot - 15}{3 \cdot 2}$

Step 5.3.1.2.3

Rewrite the expression.

$x = - \frac{- 15}{2}$

Step 5.3.2

Move the negative in front of the fraction.

$x = - - \frac{15}{2}$

Step 5.3.3

Multiply $- - \frac{15}{2}$ .

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Step 5.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{15}{2})$

Step 5.3.3.2

Multiply $\frac{15}{2}$ by $1$ .

$x = \frac{15}{2}$

Step 6

Evaluate $f (\frac{15}{2})$ .

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Step 6.1

Replace the variable $x$ with $\frac{15}{2}$ in the expression.

$f (\frac{15}{2}) = 3 {(\frac{15}{2})}^{2} - 45 (\frac{15}{2}) + 162$

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

Apply the product rule to $\frac{15}{2}$ .

$f (\frac{15}{2}) = 3 (\frac{15^{2}}{2^{2}}) - 45 (\frac{15}{2}) + 162$

Step 6.2.1.2

Raise $15$ to the power of $2$ .

$f (\frac{15}{2}) = 3 (\frac{225}{2^{2}}) - 45 (\frac{15}{2}) + 162$

Step 6.2.1.3

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

$f (\frac{15}{2}) = 3 (\frac{225}{4}) - 45 (\frac{15}{2}) + 162$

Step 6.2.1.4

Multiply $3 (\frac{225}{4})$ .

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Step 6.2.1.4.1

Combine $3$ and $\frac{225}{4}$ .

$f (\frac{15}{2}) = \frac{3 \cdot 225}{4} - 45 (\frac{15}{2}) + 162$

Step 6.2.1.4.2

Multiply $3$ by $225$ .

$f (\frac{15}{2}) = \frac{675}{4} - 45 (\frac{15}{2}) + 162$

Step 6.2.1.5

Multiply $- 45 (\frac{15}{2})$ .

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Step 6.2.1.5.1

Combine $- 45$ and $\frac{15}{2}$ .

$f (\frac{15}{2}) = \frac{675}{4} + \frac{- 45 \cdot 15}{2} + 162$

Step 6.2.1.5.2

Multiply $- 45$ by $15$ .

$f (\frac{15}{2}) = \frac{675}{4} + \frac{- 675}{2} + 162$

Step 6.2.1.6

Move the negative in front of the fraction.

$f (\frac{15}{2}) = \frac{675}{4} - \frac{675}{2} + 162$

Step 6.2.2

Find the common denominator.

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Step 6.2.2.1

Multiply $\frac{675}{2}$ by $\frac{2}{2}$ .

$f (\frac{15}{2}) = \frac{675}{4} - (\frac{675}{2} \cdot \frac{2}{2}) + 162$

Step 6.2.2.2

Multiply $\frac{675}{2}$ by $\frac{2}{2}$ .

$f (\frac{15}{2}) = \frac{675}{4} - \frac{675 \cdot 2}{2 \cdot 2} + 162$

Step 6.2.2.3

Write $162$ as a fraction with denominator $1$ .

$f (\frac{15}{2}) = \frac{675}{4} - \frac{675 \cdot 2}{2 \cdot 2} + \frac{162}{1}$

Step 6.2.2.4

Multiply $\frac{162}{1}$ by $\frac{4}{4}$ .

$f (\frac{15}{2}) = \frac{675}{4} - \frac{675 \cdot 2}{2 \cdot 2} + \frac{162}{1} \cdot \frac{4}{4}$

Step 6.2.2.5

Multiply $\frac{162}{1}$ by $\frac{4}{4}$ .

$f (\frac{15}{2}) = \frac{675}{4} - \frac{675 \cdot 2}{2 \cdot 2} + \frac{162 \cdot 4}{4}$

Step 6.2.2.6

Multiply $2$ by $2$ .

$f (\frac{15}{2}) = \frac{675}{4} - \frac{675 \cdot 2}{4} + \frac{162 \cdot 4}{4}$

Step 6.2.3

Combine the numerators over the common denominator.

$f (\frac{15}{2}) = \frac{675 - 675 \cdot 2 + 162 \cdot 4}{4}$

Step 6.2.4

Simplify each term.

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Step 6.2.4.1

Multiply $- 675$ by $2$ .

$f (\frac{15}{2}) = \frac{675 - 1350 + 162 \cdot 4}{4}$

Step 6.2.4.2

Multiply $162$ by $4$ .

$f (\frac{15}{2}) = \frac{675 - 1350 + 648}{4}$

Step 6.2.5

Simplify the expression.

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Step 6.2.5.1

Subtract $1350$ from $675$ .

$f (\frac{15}{2}) = \frac{- 675 + 648}{4}$

Step 6.2.5.2

Add $- 675$ and $648$ .

$f (\frac{15}{2}) = \frac{- 27}{4}$

Step 6.2.5.3

Move the negative in front of the fraction.

$f (\frac{15}{2}) = - \frac{27}{4}$

Step 6.2.6

The final answer is $- \frac{27}{4}$ .

$- \frac{27}{4}$

Step 7

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

$(\frac{15}{2}, - \frac{27}{4})$

Step 8