Finding critical points. Then find and classifying all the extreme values...?

Question

Find the critical points. Then find and classify all the extreme values:

(I) f(x) = 2x^2 +5x -1, x E [-2,0]

(II) f(x) = x^2 / (1+x^2), x E [-1,2]

Where E is that sideways looking M, not sigma (unless they mean the same thing?).

(III) f(x) = sin2x - x, 0<=x<=pi

Explain your work. Thanks! :)

Answer 1

I'll do (i) and then you can do the others by example.

f(x) = 2x^2 + 5x - 1 for -2 <=x <= 0 (that symbol means for all x contained on the closed interval (the square brackets] -2 to 0)

To find the critical points, take the first derivative and set it equal to zero:

df/dx = 4x + 5 = 0
Solve for the value of x that make df/dx = 0 ---> x =-5/4 = -1.25

Now to determine what kind of critical point, take the second derivative and evaluate it at x = -1.25

d^2f/dx^2 = 4 since this is positive, the point at x=-1.25 is a minimum.