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Putnam 1967 B6

Source: Putnam 1967

May 14, 2022
Putnamfunctioncalculusderivativepartial derivatives

Problem Statement

Let ff be a real-valued function having partial derivatives and which is defined for x2+y2≤1x^2 +y^2 \leq1 and is such that ∣f(x,y)∣≤1.|f(x,y)|\leq 1. Show that there exists a point (x0,y0)(x_0, y_0 ) in the interior of the unit circle such that (∂f∂x(x0,y0))2+(∂f∂y(x0,y0))2≤16.\left( \frac{ \partial f}{\partial x}(x_0 ,y_0 ) \right)^{2}+ \left( \frac{ \partial f}{\partial y}(x_0 ,y_0 ) \right)^{2} \leq 16.
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