Analysis | Part IA, 2005

Explain what it means for a bounded function f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} to be Riemann integrable.

Let f:[0,)Rf:[0, \infty) \rightarrow \mathbb{R} be a strictly decreasing continuous function. Show that for each x(0,)x \in(0, \infty), there exists a unique point g(x)(0,x)g(x) \in(0, x) such that

1x0xf(t)dt=f(g(x)).\frac{1}{x} \int_{0}^{x} f(t) d t=f(g(x)) .

Find g(x)g(x) if f(x)=exf(x)=e^{-x}.

Suppose now that ff is differentiable and f(x)<0f^{\prime}(x)<0 for all x(0,)x \in(0, \infty). Prove that gg is differentiable at all x(0,)x \in(0, \infty) and g(x)>0g^{\prime}(x)>0 for all x(0,)x \in(0, \infty), stating clearly any results on the inverse of ff you use.

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