If f is a continuous function of one variable with tworelative maxima on an interval

Chapter 13, Problem 30

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Focus on Concepts

If \(f\) is a continuous function of one variable with two relative maxima on an interval, then there must be a relative minimum between the relative maxima. (Convince yourself of this by drawing some pictures.) The purpose of this exercise is to show that this result does not extend to functions of two variables. Show that \(f(x, y)=4 x^{2} e^{y}-2 x^{4}-e^{4 y}\) has two relative maxima but no other critical points (see Figure Ex-30).

Source: This exercise is based on the problem “Two Mountains Without a Valley” proposed and solved by Ira Rosenholtz, Mathematics Magazine, Vol. 60, No. 1, February 1987, p. 48.

                                         

Equation Transcription:

Text Transcription:

f

f(x,y)=4x^2e^y-2x^4-e^4y