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- 807

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## Homework Statement

Given f(t) = integral {(x^2 + 12x + 35)/(1 + cos^2 (x)) dx}.

At what value of t does the local max f(t) occur?

(A more aesthetically-pleasing version of the question is attached as TheQuestion.jpg.)

## Homework Equations

Differentiating (I think).

## The Attempt at a Solution

I know I can get f'(t) by just replacing the “x”s with “t”s and removing the integral because of the way the limits of integration are set up and, differentiating and integrating undo each other.

I also know that I can find minima and maxima using f'(t) = 0 but, that gives me two answers t = –7 or t = –5 and, I need to know which one it is.

Am I expected to find f''(t) = 0 to see which is a minimum and which is a maximum? (I ask because that seems quite complex.)

Or, am I supposed to just pick values of t slightly larger and slightly smaller than each value of t that I have to determine this? (This doesn't sit well with me because, while it might work well in practice a lot of the time, it isn't theoretically fool-proof.)

Any input would be greatly appreciated!