How Do You Solve This Fundamental Theorem Problem?

[Alem2000]

My professer told us to think about this problem. I have the answer in my solutions manual but I want to know how I would go through the thinking process to solve it...(and others like it) The question says

"Find a function f and a number a such that

$$6+\int_{a}^{x}\frac{f(t)}{t^2}dt=2\sqrt{x}$$
for all x greater than zero"

 

[Tide]

Differentiate the equation!

 

[Alem2000]

There was no point in that reply..
Anyone out there that can really help?

 

[Tide]

There was no point in that reply..
Anyone out there that can really help?

Begging your pardon but I told you exactly what you need to do!

 

[Muzza]

There was no point in that reply..

WTF? Tide's post was immensely helpful and practically gives you the entire solution.

 

[Gokul43201]

I second that. Tide's given you a correct way to solve the problem.

 

[HallsofIvy]

Since you did not grasp what Tide said, here's another way of looking at it:
Your equation is equivalent to
$$\int_{a}^{x}\frac{f(t)}{t^2}dt=2\sqrt{x}- 6$$


Do you notice that the right hand side is a constant?

 

[Muzza]

The right hand side is not a constant...

 

[HallsofIvy]

The right hand side is not a constant...

OMIGOD! I stared at that repeatedly and saw $2\sqrt{2}$.

 

[HallsofIvy]

Alem2000: as Tide said, differentiate both sides:
The derivative of $$\int_{a}^{x}\frac{f(t)}{t^2}dt$$ is $$\frac{f(x)}{x^2}$$ (that's the "fundamental theorem your title referred to) and the derivative of $$2\sqrt{2}= 2(x^{1/2})$$ is $$x^{-1/2}$$.

Set them equal and solve for x.

 

[Alem2000]

OOOOOO! I think I made that way more complicated then it was. Thanks a lot Tide...sorry about the frustration

 

[HallsofIvy]

I did it again! I wrote $$2\sqrt{2}$$ when I meant $$2\sqrt{x}$$!

 

[JonF]

And now you are throwing factorial signs about, tisk tisk ;)