MathDB

CIMA 2014 Problem 6

Source:

August 23, 2014

functionintegrationcollege contestsreal analysis

Problem Statement

(a) Show that if

f:[-1,1]\to \mathbb{R}

is a convex and

C^2

function such that

f(1),f(-1)\geq 0

, then:

\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''

(b) Let

B\subset \mathbb{R}^2

the closed ball with center

0

and radius

1

. Show that if

f: B \to \mathbb{R}

is a convex and

C^2

function and

f\geq 0

in

\partial B

, then:

f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}

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