Calculus Examples

$f (x) = a x^{2} + 7 x - 3$

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 $x = - \frac{b}{2 a}$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Remove parentheses.

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

Step 2.3

Multiply $2$ by $1$ .

$x = - \frac{7}{2}$

Step 3

Evaluate $f (- \frac{7}{2})$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $- \frac{7}{2}$ in the expression.

$f (- \frac{7}{2}) = a {(- \frac{7}{2})}^{2} + 7 (- \frac{7}{2}) - 3$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

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

Tap for more steps...

Step 3.2.1.1.1

Apply the product rule to $- \frac{7}{2}$ .

$f (- \frac{7}{2}) = a ({(- 1)}^{2} {(\frac{7}{2})}^{2}) + 7 (- \frac{7}{2}) - 3$

Step 3.2.1.1.2

Apply the product rule to $\frac{7}{2}$ .

$f (- \frac{7}{2}) = a ({(- 1)}^{2} (\frac{7^{2}}{2^{2}})) + 7 (- \frac{7}{2}) - 3$

Step 3.2.1.2

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

$f (- \frac{7}{2}) = a (1 (\frac{7^{2}}{2^{2}})) + 7 (- \frac{7}{2}) - 3$

Step 3.2.1.3

Multiply $\frac{7^{2}}{2^{2}}$ by $1$ .

$f (- \frac{7}{2}) = a (\frac{7^{2}}{2^{2}}) + 7 (- \frac{7}{2}) - 3$

Step 3.2.1.4

Raise $7$ to the power of $2$ .

$f (- \frac{7}{2}) = a (\frac{49}{2^{2}}) + 7 (- \frac{7}{2}) - 3$

Step 3.2.1.5

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

$f (- \frac{7}{2}) = a (\frac{49}{4}) + 7 (- \frac{7}{2}) - 3$

Step 3.2.1.6

Combine $a$ and $\frac{49}{4}$ .

$f (- \frac{7}{2}) = \frac{a \cdot 49}{4} + 7 (- \frac{7}{2}) - 3$

Step 3.2.1.7

Move $49$ to the left of $a$ .

$f (- \frac{7}{2}) = \frac{49 \cdot a}{4} + 7 (- \frac{7}{2}) - 3$

Step 3.2.1.8

Multiply $7 (- \frac{7}{2})$ .

Tap for more steps...

Step 3.2.1.8.1

Multiply $- 1$ by $7$ .

$f (- \frac{7}{2}) = \frac{49 a}{4} - 7 (\frac{7}{2}) - 3$

Step 3.2.1.8.2

Combine $- 7$ and $\frac{7}{2}$ .

$f (- \frac{7}{2}) = \frac{49 a}{4} + \frac{- 7 \cdot 7}{2} - 3$

Step 3.2.1.8.3

Multiply $- 7$ by $7$ .

$f (- \frac{7}{2}) = \frac{49 a}{4} + \frac{- 49}{2} - 3$

Step 3.2.1.9

Move the negative in front of the fraction.

$f (- \frac{7}{2}) = \frac{49 a}{4} - \frac{49}{2} - 3$

Step 3.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{7}{2}) = \frac{49 a}{4} - \frac{49}{2} - 3 \cdot \frac{2}{2}$

Step 3.2.3

Combine $- 3$ and $\frac{2}{2}$ .

$f (- \frac{7}{2}) = \frac{49 a}{4} - \frac{49}{2} + \frac{- 3 \cdot 2}{2}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (- \frac{7}{2}) = \frac{49 a}{4} + \frac{- 49 - 3 \cdot 2}{2}$

Step 3.2.5

Simplify the numerator.

Tap for more steps...

Step 3.2.5.1

Multiply $- 3$ by $2$ .

$f (- \frac{7}{2}) = \frac{49 a}{4} + \frac{- 49 - 6}{2}$

Step 3.2.5.2

Subtract $6$ from $- 49$ .

$f (- \frac{7}{2}) = \frac{49 a}{4} + \frac{- 55}{2}$

Step 3.2.6

Move the negative in front of the fraction.

$f (- \frac{7}{2}) = \frac{49 a}{4} - \frac{55}{2}$

Step 3.2.7

The final answer is $\frac{49 a}{4} - \frac{55}{2}$ .

$\frac{49 a}{4} - \frac{55}{2}$

Step 4

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

$(- \frac{7}{2}, \frac{49 a}{4} - \frac{55}{2})$

Step 5