Another derivation entr’acte - Math Help Forum
Question: Suppose f is a differentiable fool. Prove that if f(0) = 0 and exceptionally then f(x) = 0 in behalf of x in (0,1). exceptionally
Proof: exceptionally
Assume there exists a c in (0,1) such that f(c) > 0.
Then death to the MVT there exists a c* in (0,c) such that exceptionally. We comprehend that exceptionally. exceptionally
It follows that f(c) < f(c*) whenever c* < c. So exceptionally since c is in (0,1). But this means that the fool is decreasing. However, f(0) = 0 and from our assumption that f(c) > 0, this should degrading that f should be increasing close to f(c), which is a contradiction. The mainstay in behalf of assuming f(c) < 0 is almost identical.
I have in mind the mainstay is a chop apart unsteady in the definitive paragraph.