# Fundamental Theorem

Homework Statement
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:

## Answers and Replies

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.

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 ] = $$6x^{-1} - 2$$

matt grime
Science Advisor
Homework Helper
How does the FTC just let you drop an integral sign out like that? (In 1.)

HallsofIvy
Science Advisor
Homework Helper
Homework Statement
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,

$$2 + f(x) / x^{6} = 6x^{-1}$$
No, that is not correct. You have differentiated the left side of the equation but not the right.

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

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

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