Calculus Examples

$f (x) = a x^{2} + b x + c$

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 $b$ .

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

Step 2.2

Remove parentheses.

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

Step 2.3

Cancel the common factor of $1$ .

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Step 2.3.1

Cancel the common factor.

$x = - \frac{1}{2 \cdot 1}$

Step 2.3.2

Rewrite the expression.

$x = - \frac{1}{2}$

Step 3

Evaluate $f (- \frac{1}{2})$ .

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Step 3.1

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

$f (- \frac{1}{2}) = a {(- \frac{1}{2})}^{2} + b (- \frac{1}{2}) + c$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

$f (- \frac{1}{2}) = a ({(- 1)}^{2} {(\frac{1}{2})}^{2}) + b (- \frac{1}{2}) + c$

Step 3.2.1.1.2

Apply the product rule to $\frac{1}{2}$ .

$f (- \frac{1}{2}) = a ({(- 1)}^{2} (\frac{1^{2}}{2^{2}})) + b (- \frac{1}{2}) + c$

Step 3.2.1.2

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

$f (- \frac{1}{2}) = a (1 (\frac{1^{2}}{2^{2}})) + b (- \frac{1}{2}) + c$

Step 3.2.1.3

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

$f (- \frac{1}{2}) = a (\frac{1^{2}}{2^{2}}) + b (- \frac{1}{2}) + c$

Step 3.2.1.4

One to any power is one.

$f (- \frac{1}{2}) = a (\frac{1}{2^{2}}) + b (- \frac{1}{2}) + c$

Step 3.2.1.5

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

$f (- \frac{1}{2}) = a (\frac{1}{4}) + b (- \frac{1}{2}) + c$

Step 3.2.1.6

Combine $a$ and $\frac{1}{4}$ .

$f (- \frac{1}{2}) = \frac{a}{4} + b (- \frac{1}{2}) + c$

Step 3.2.1.7

Combine $b$ and $\frac{1}{2}$ .

$f (- \frac{1}{2}) = \frac{a}{4} - \frac{b}{2} + c$

Step 3.2.2

The final answer is $\frac{a}{4} - \frac{b}{2} + c$ .

$\frac{a}{4} - \frac{b}{2} + c$

Step 4

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

$(- \frac{1}{2}, \frac{a}{4} - \frac{b}{2} + c)$

Step 5