Proving Rolle's Theorem: Proving Zero's of f(x) in [0,1]

[nlews]

This is the last part of a problem that I'm working through. The problem is on Rolle's theorem.

Using Rolles Theorem prove that for any real number λ: the function f(x) = x^3 - (3/2)x^2 + λ never has two distinct zeros in [0,1].

So I was thinking about ways I could do this: but when I calculated f(0) = λ, wheras f(1) = λ-1/2, but to use Rolle's theorem isn't f(a) = f(b) on the interval [a,b]?? This has got me a little confused.
Anyway I put that aside to try another way:
I thought perhaps I should assume for contradiction that I should assume there are two distinct zeros at c1 and c2. So f'(c1 = f'(c2) = 0. But then using Rolles Theorem there is a root of f between c1 and c2. But now I don't know where to go with this next! Please help!

 

[elibj123]

Rolle's theorem states that if a function is identical on two distinct points between which it's defined and has a derivative, then there is a point between them where the derivative vanishes.
So after making the assumption, you have f(c1)=f(c2)=0 rather than f'(c1)=f'(c2)=0 (as you wrote), which means there is a point c where f'(c)=0, and 0<c<1. Is this possible?

 

[Redbelly98]

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