- #1

Yagoda

- 46

- 0

## Homework Statement

Suppose that f is continuous on [a,b], its right-hand derivative [itex]f'_{+}(a)[/itex] exists, and [itex]\mu[/itex] is between [itex]f'_{+}(a)[/itex] and [itex]\frac{f(b)-f(a)}{b-a}[/itex]. Show that [itex]f(c) - f(a) = \mu (c-a)[/itex] for some [itex]c \in (a,b)[/itex].

## Homework Equations

[itex] f'_{+}(a) = \lim_{x \rightarrow a^+}\frac{f(x)-f(a)}{x-a} [/itex]

## The Attempt at a Solution

I spent a little while trying to figure out exactly what I am trying to prove and it definitely makes sense now. Intuitively I can see that it is true. My first instinct was to use the IVT for derivatives or the MVT, but those both require that f be differentiable, which I don't know to be the case here.

So I looked at using the regular IVT, since it only requires that f be continuous on [a,b], but also needs f(a) not equal to f(b). I was going to for now ignore the case where f(a) = f(b), but I was having trouble showing that [itex]\mu [/itex] is between the required bounds.

Am I even on the right track or is there another approach that makes more sense?

Thanks.