Calculus Examples

$2 - 7 x^{2}$

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

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

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

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

$x = - \frac{0}{2 (- 7)}$

Remove parentheses.

$x = - \frac{0}{2 (- 7)}$

Simplify $- \frac{0}{2 (- 7)}$ .

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Cancel the common factor of $0$ and $2$ .

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Factor $2$ out of $0$ .

$x = - \frac{2 (0)}{2 (- 7)}$

Cancel the common factors.

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Cancel the common factor.

$x = - \frac{2 \cdot 0}{2 \cdot - 7}$

Rewrite the expression.

$x = - \frac{0}{- 7}$

Cancel the common factor of $0$ and $- 7$ .

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Factor $7$ out of $0$ .

$x = - \frac{7 (0)}{- 7}$

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$x = - (- 1 \cdot 0)$

Simplify the expression.

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Rewrite $- 1 \cdot 0$ as $- 0$ .

$x = - - 0$

Multiply $- 1$ by $0$ .

$x = - 0$

Multiply $- 1$ by $0$ .

$x = 0$

Evaluate $f (0)$ .

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Replace the variable $x$ with $0$ in the expression.

$f (0) = 2 - 7 {(0)}^{2}$

Simplify the result.

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Simplify each term.

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Raising $0$ to any positive power yields $0$ .

$f (0) = 2 - 7 \cdot 0$

Multiply $- 7$ by $0$ .

$f (0) = 2 + 0$

Add $2$ and $0$ .

$f (0) = 2$

The final answer is $2$ .

$2$

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

$(0, 2)$