Paper 1, Section II, F

Analysis I | Part IA, 2021

Let f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} be a continuous function.

(a) Let m=minx[a,b]f(x)m=\min _{x \in[a, b]} f(x) and M=maxx[a,b]f(x)M=\max _{x \in[a, b]} f(x). If g:[a,b]Rg:[a, b] \rightarrow \mathbb{R} is a positive continuous function, prove that

mabg(x)dxabf(x)g(x)dxMabg(x)dxm \int_{a}^{b} g(x) d x \leqslant \int_{a}^{b} f(x) g(x) d x \leqslant M \int_{a}^{b} g(x) d x

directly from the definition of the Riemann integral.

(b) Let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} be a continuous function. Show that

01/nnf(x)enxdxf(0)\int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x \rightarrow f(0)

as nn \rightarrow \infty, and deduce that

01nf(x)enxdxf(0)\int_{0}^{1} n f(x) e^{-n x} d x \rightarrow f(0)

as nn \rightarrow \infty

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