Calculus Examples

$f (x) = 7 x^{2} + x$

Step 1

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 2

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

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Step 2.1

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

$x = - \frac{1}{2 (7)}$

Step 2.2

Remove parentheses.

$x = - \frac{1}{2 (7)}$

Step 2.3

Multiply $2$ by $7$ .

$x = - \frac{1}{14}$

Step 3

Evaluate $f (- \frac{1}{14})$ .

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Step 3.1

Replace the variable $x$ with $- \frac{1}{14}$ in the expression.

$f (- \frac{1}{14}) = 7 {(- \frac{1}{14})}^{2} - \frac{1}{14}$

Step 3.2

Simplify the result.

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Step 3.2.1

Remove parentheses.

$f (- \frac{1}{14}) = 7 {(- \frac{1}{14})}^{2} - \frac{1}{14}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.2.1.1

Apply the product rule to $- \frac{1}{14}$ .

$f (- \frac{1}{14}) = 7 ({(- 1)}^{2} {(\frac{1}{14})}^{2}) - \frac{1}{14}$

Step 3.2.2.1.2

Apply the product rule to $\frac{1}{14}$ .

$f (- \frac{1}{14}) = 7 ({(- 1)}^{2} (\frac{1^{2}}{14^{2}})) - \frac{1}{14}$

Step 3.2.2.2

Raise $- 1$ to the power of $2$ .

$f (- \frac{1}{14}) = 7 (1 (\frac{1^{2}}{14^{2}})) - \frac{1}{14}$

Step 3.2.2.3

Multiply $\frac{1^{2}}{14^{2}}$ by $1$ .

$f (- \frac{1}{14}) = 7 (\frac{1^{2}}{14^{2}}) - \frac{1}{14}$

Step 3.2.2.4

One to any power is one.

$f (- \frac{1}{14}) = 7 (\frac{1}{14^{2}}) - \frac{1}{14}$

Step 3.2.2.5

Raise $14$ to the power of $2$ .

$f (- \frac{1}{14}) = 7 (\frac{1}{196}) - \frac{1}{14}$

Step 3.2.2.6

Cancel the common factor of $7$ .

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Step 3.2.2.6.1

Factor $7$ out of $196$ .

$f (- \frac{1}{14}) = 7 (\frac{1}{7 (28)}) - \frac{1}{14}$

Step 3.2.2.6.2

Cancel the common factor.

$f (- \frac{1}{14}) = 7 (\frac{1}{7 \cdot 28}) - \frac{1}{14}$

Step 3.2.2.6.3

Rewrite the expression.

$f (- \frac{1}{14}) = \frac{1}{28} - \frac{1}{14}$

Step 3.2.3

To write $- \frac{1}{14}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{1}{14}) = \frac{1}{28} - \frac{1}{14} \cdot \frac{2}{2}$

Step 3.2.4

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.4.1

Multiply $\frac{1}{14}$ by $\frac{2}{2}$ .

$f (- \frac{1}{14}) = \frac{1}{28} - \frac{2}{14 \cdot 2}$

Step 3.2.4.2

Multiply $14$ by $2$ .

$f (- \frac{1}{14}) = \frac{1}{28} - \frac{2}{28}$

Step 3.2.5

Combine the numerators over the common denominator.

$f (- \frac{1}{14}) = \frac{1 - 2}{28}$

Step 3.2.6

Subtract $2$ from $1$ .

$f (- \frac{1}{14}) = \frac{- 1}{28}$

Step 3.2.7

Move the negative in front of the fraction.

$f (- \frac{1}{14}) = - \frac{1}{28}$

Step 3.2.8

The final answer is $- \frac{1}{28}$ .

$- \frac{1}{28}$

Step 4

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

$(- \frac{1}{14}, - \frac{1}{28})$

Step 5