Question

Find and classify the relative maxima and minima of this function
f(x)= definite integral sign where a=0 and b=x.

t^2-4/(1+(cos^2(t)) dt

what should my value for u be? is it just t. not sure how to even tackle this problem.

Answer 1

The clue here is that we want to find extrema of f(x), which involves finding the derivative of f. Since f(x) is defined as an integral, we don't really have to do the integration. We just apply the rules for differentiating under the integral sign. (See wikipedia, and scroll down for some examples)

f(x) = ∫[0,x] (t^2-4)/(1+(cos^2(t)) dt

so,

df/dx = (x^2-4)/(1+cos^2 x)

So, f'=0 when x^2-4 = 0, since the bottom is never zero.

Obviously the extrema are at x=2,-2.

Answer 2

Thank you!