Mean Value Theorem problem, where did I go wrong?

[ECmathstudent]

Homework Statement

Suppose that f is continuous on [a,b], and dy/dx f(a+)< u < (f(b) - f(a))/(b-a) that there exists a point c so that u(c-a) = f(c)-f(a)

Homework Equations

The Mean Value Theorem, Intermediate value theorem

The Attempt at a Solution

I defined (f(b) - f(a))/(b-a) = dy/dx f(d) for some d in [a,b]
then set the point n in [a,b] so I could state
f(b) = (dy/dx f(d))(b-a) + f(a)
f(n) = (dy/dx f(a))(b-a) + f(a)

so with no loss in generality I put the inequalities in this order

(dy/dx f(a))(b-a) + f(a) < u(b-a) + f(a) < (dy/dx f(a))(b-a) + f(a)
so
f(n) < u(b-a) + f(a) < f(b)

so by Intermediate value theorem there is some point p in [n,b] so that

f(p) = u(b-a) + f(a)
so
f(p) - f(a) = u(b-a)
which is not what I wanted, any idea what I could've done to make that b a p in the final step, or should I start from scratch?

 

[ECmathstudent]

And I just realized I missed the day where we learned Darboux's theorem, and spent 4 hours working on problems trying to derive it on my own.