# Fundamental Theorem

1. Jul 13, 2008

### irok

The problem statement, all variables and given/known data
Question One:
Find a continuous function f and a number a such that

$$2 + \int_{a}^{x} \frac {f(t)} {t^{6}} \,dt = 6 x^{-1}$$

Question Two:
At what value of x does the local max of f(x) occur?
$$f(x) = \int_0^x \frac{ t^2 - 25 }{ 1+\cos^2(t)} dt$$

The attempt at a solution
I just need some pointers of where to get started.
Question One:

So I used FTC1 on both sides,

$$2 + f(x) / x^{6} = 6x^{-1}$$

$$f(x)= 6x^{5} - 2$$

I'm not sure how to find a, evaluation theorem?

Question Two:

Last edited: Jul 13, 2008
2. Jul 13, 2008

I don't know about an analytic solution, but the second part of the problem is very feasible numerically. You can solve in Mathematica in only a few lines by turning it into a minimization problem.

3. Jul 13, 2008

### irok

Well, for Question one:

Can anyone confirm that $$f(x) = 6x^{5}$$ and a = 2.

I'm pretty sure that a = 2 since,

F(x) - F(a) = [ $$6x^{5} / x^{6}$$ ] - [ 2 ] = $$6x^{-1} - 2$$

4. Jul 13, 2008

### matt grime

How does the FTC just let you drop an integral sign out like that? (In 1.)

5. Jul 14, 2008

### HallsofIvy

Staff Emeritus
No, that is not correct. You have differentiated the left side of the equation but not the right.

Once you have found the function, put it into the integeral.