Question

Given f(x)=∫ (t^2−1)/(1+cos^2(t))dt At what value of x does the local max of f(x) occur ? (a=0 and b =x)

Answer 1

Use the Fundamental Theorem of Calculus to find `f'(x)`. Then find critical numbers and finally, use the first derivative test to determine where the local maximum occurs.

Explanation:

Given: `f(x) = int_0^x (t^2-1)/(1+cos^2t)dt`

I assume that the domain is as large as possible, namely `RR`.

Part 1 of the Fundamental Theorem of Calculus tells us that

`f'(x) = (x^2-1)/(1+cos^2x)`.

This derivative is never undefined and is `0` at `x=+-1`.

Therefore, the critical numbers for `f` are `-1` and `1`.

The denominator of `f'` is always positive, so
the sign of `f'` is the same as the sign of `x^2-1` which is

positive on `(-oo,-1)` and `(1,oo)`, and
negative on `(-1,1)`.

`f'` changes from + to - at `-1`, so `f(-1)` is a local maximum.