1.II.11C

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

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

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

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

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

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

Find the solution of the equation

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

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

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