# How do you find the maximum value of y = −2x^2 − 3x + 2?

Jul 14, 2016

The maximum value of the function is $\frac{25}{8}$.

#### Explanation:

We can tell two things about this function before we begin approaching the problem:

1) As $x \to - \infty$ or $x \to \infty$, $y \to - \infty$. This means our function will have an absolute maximum, as opposed to a local maximum or no maxima at all.

2) The polynomial is of degree two, meaning it changes direction only once. Thus, the only point at which is changes direction must also be our maximum. In a higher degree polynomial, it might be necessary to compute multiple local maxima and determine which is the largest.

To find the maximum, we first find the $x$ value at which the function changes direction. this will be the point where $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$.
$\frac{\mathrm{dy}}{\mathrm{dx}} = - 4 x - 3$
$0 = - 4 x - 3$
$3 = - 4 x$
$x = - \frac{3}{4}$

This point must be our local maximum. The value at that point is determined by calculating the value of the function at that point:
$y = - 2 {\left(- \frac{3}{4}\right)}^{2} - 3 \left(- \frac{3}{4}\right) + 2$
$= - \frac{18}{16} + \frac{9}{4} + 2$
$= - \frac{9}{8} + \frac{18}{8} + \frac{16}{8}$
$= \frac{25}{8}$