Critical points AND extreme values...?

Question

Find the critical points, then find and classify all the extreme values in:

f(x) = 2x^2 +5x -1, for x in the interval [-2,0]

I have about 5 problems similar to this, but if you can show me how to do all the steps in this one, I think I can do the rest. Thanks :)

Answer 1

f´(x) = 4x+5 =0 so x= -5/4 inside the interval(sign-----(-5/4)++++
sign of f(x)
2x^2+5x-1=0 x= ((-5+-sqrt(25+8))/4
sign +++++(-5-sqrt(33)/4 ----- (-2) ------0+++ (-5+sqrt(33))/4
f(-5/4) is a relative minimum and also the absolute one
f(0) is the absolute maximum

Answer 2

Extreme values refer to the absolute maximum and absolute minimum of f(x).

To find extreme values, we use the closed interval method:

so, for this question, end points: -2, 0

Critical points are the points where f'(x)=0 or f'(x) does not exist.

f'(x) = 4x + 5
when f'(x) = 0, x= - 5/4

f'(x) exists for all real values of x.

critical point: -5/4

Compare the values of f(x) at the end points and the critical points:

f(-2) = 8-10-1 = -3
f(0) = -1
f(-5/4) = 25/8 -25/4 -1 = -33/8

Thus, absolute max: -1
absolute min: -33/8