Analysis I | Part IA, 2002

State the mean value theorem and deduce it from Rolle's theorem.

Use the mean value theorem to show that, if h:RRh: \mathbb{R} \rightarrow \mathbb{R} is differentiable with h(x)=0h^{\prime}(x)=0 for all xx, then hh is constant.

By considering the derivative of the function gg given by g(x)=eaxf(x)g(x)=e^{-a x} f(x), find all the solutions of the differential equation f(x)=af(x)f^{\prime}(x)=a f(x) where f:RRf: \mathbb{R} \rightarrow \mathbb{R} is differentiable and aa is a fixed real number.

Show that, if f:RRf: \mathbb{R} \rightarrow \mathbb{R} is continuous, then the function F:RRF: \mathbb{R} \rightarrow \mathbb{R} given by

F(x)=0xf(t)dtF(x)=\int_{0}^{x} f(t) d t

is differentiable with F(x)=f(x)F^{\prime}(x)=f(x).

Find the solution of the equation

g(x)=A+0xg(t)dtg(x)=A+\int_{0}^{x} g(t) d t

where g:RRg: \mathbb{R} \rightarrow \mathbb{R} is differentiable and AA is a real number. You should explain why the solution is unique.

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