Proving the Derivative Function for a Differentiable Function on an Interval

[tdwp]

Homework Statement

Let f be a differentiable function defined for all x>=0 such that f(0)=5 and f(3)=-1. Suppose that for any number b>0, the average value of f(x) on the interval 0<=x<=b is (f(0)+f(b))/x

a. Find the integral of f(x) from 0 to 3.
b. Prove that f'(x)=(f(x)-5)/x for all x<0.
c. Using part b), find f(x)

Homework Equations

(b-a)(f((ave)x))= the integral of f(x) from a to b

The Attempt at a Solution

Part a is easy, I got 6 as my answer. I'm completely at a loss on how to do part b/c. If anyone would at least point me in the right direction, I would greatly appreciate it.

 

[AKG]

Suppose that for any number b>0, the average value of f(x) on the interval 0<=x<=b is (f(0)+f(b))/x

Either you copied out the question wrong or the question asked doesn't make sense.

 

[tdwp]

Argh, yes I did copy it wrong. It should be (f(0)+f(b))/2.

 

[AKG]

Use your "relevant equations" together with the given formula for the average of f over [0,b] to get an equation. Differentiate both sides of the equation.