Paper 1, Section II, 10F10 F

Analysis I | Part IA, 2018

(a) Let f:RRf: \mathbb{R} \rightarrow \mathbb{R} be differentiable at x0Rx_{0} \in \mathbb{R}. Show that ff is continuous at x0x_{0}.

(b) State the Mean Value Theorem. Prove the following inequalities:

cos(ex)cos(ey)xy for x,y0\left|\cos \left(e^{-x}\right)-\cos \left(e^{-y}\right)\right| \leqslant|x-y| \quad \text { for } x, y \geqslant 0


log(1+x)x1+x for x0.\log (1+x) \leqslant \frac{x}{\sqrt{1+x}} \text { for } x \geqslant 0 .

(c) Determine at which points the following functions from R\mathbb{R} to R\mathbb{R} are differentiable, and find their derivatives at the points at which they are differentiable:

f(x)={xx if x01 if x=0,g(x)=cos(x),h(x)=xxf(x)=\left\{\begin{array}{ll} |x|^{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0, \end{array} \quad g(x)=\cos (|x|), \quad h(x)=x|x|\right.

(d) Determine the points at which the following function from R\mathbb{R} to R\mathbb{R} is continuous:

f(x)={0 if xQ or x=01/q if x=p/q where pZ\{0} and qN are relatively prime. f(x)= \begin{cases}0 & \text { if } x \notin \mathbb{Q} \text { or } x=0 \\ 1 / q & \text { if } x=p / q \text { where } p \in \mathbb{Z} \backslash\{0\} \text { and } q \in \mathbb{N} \text { are relatively prime. }\end{cases}

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