Maxima/Minima Question

[science.girl]

1. The problem statement, all variables and given/known data
Find relative extrema of f(x).

f(x) = \int^{x}_{0} (t^{2} -4)/(1 + cos(t)^{2})


2. Relevant equations
N/A


3. The attempt at a solution
Is this correct?

f '(x) = [(x² - 4)/(1 + cos²x)]

Now set f '(x) = 0,
[(x² - 4)/(1 + cos²x)] = 0
x² - 4 = 0
x = ± 2

f'(x) is changing from negative to positive for both +2 and -2, so are both of them minima?
(And would you have to take the derivative of the so-called f'(x) to get the actual derivative from which to calculate the max/min?)

[ideasrule]

Your work is right; deriving an integral yields the original equation inside the integral.

[science.girl]

Your work is right; deriving an integral yields the original equation inside the integral.

Thank you for the clarification. =)

[jbunniii]

f'(x) is changing from negative to positive for both +2 and -2

No, that's not true. f'(x) is an even function of x, so whatever its slope is at -2 must be the negative of its slope at 2.

[science.girl]

No, that's not true. f'(x) is an even function of x, so whatever its slope is at -2 must be the negative of its slope at 2.

Ah; makes sense. Thank you for pointing this out!