Extreme point on a parabola:

 Find the minimum for the quadratic
.

First, draw a graph of this parabola.

Visually, we can determine that the minimum is between -2 and 5.

The minimum is the extreme point and can be found by setting the derivative to 0 and solving for
.

The derivative is
.

Carrying out the differentiation, we get
.

At the (unknown) value, , we know that the derivative is 0:

Substituting from above, we get

.

Solving for
,
we get
,

Notice that this agrees with the graph in that the minimum appears to be midway between the points -2 and 5 where the curve crosses the x axis. This is generally true for quadratics, but not for other polynomials.

We can find the y coordinate of this point by using

This agrees with the graph.

Test Problem: Find the x and y coordinates of the maximum for

 Find the x coordinate of the maximum:

a) -1

b) 1/2

c) -1/2

d) 5/2

Find the y coordinate of the maximum:

e) 6.25

f) 6.75

g) -6.25

h) 2.25