Recall from Theorem 4.4.4 that if a continuous functionof one variable has exactly one

Chapter 13, Problem 29

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Focus on Concepts

Recall from Theorem 4.4.4 that if a continuous function of one variable has exactly one relative extremum on an interval, then that relative extremum is an absolute extremum on the interval. This exercise shows that this result does not extend to functions of two variables.

(a) Show that \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) has only one critical point and that a relative maximum occurs there. (See the accompanying figure.)

(b) Show that \(f\) does not have an absolute maximum.

                                               

Source: This exercise is based on the article “The Only Critical Point in Town Test” by Ira Rosenholtz and Lowell Smylie, Mathematics Magazine, Vol. 58, No. 3, May 1985, pp. 149–150.

Equation Transcription:

Text Transcription:

f(x,y)=3xe^y-x^3-e3^y

f