Question

Find the local maximum and minimum of f(x,y)=y(e^x−1):

A. No local maximum or minimum
B. Local maximum at y=0 and no local minimum
C. Local minimum at y=0 and no local maximum
D. Local maximum at (0,0) and no local minimum

Answer 1

The function f(x,y)=y(e^x-1) has no local maximum or minimum since y=0 is a critical line, but the function does not reach a maximum or minimum for any specific x value.

Step-by-step explanation:

To find the local maximum and minimum of the function f(x,y)=y(e^x-1), we need to calculate the partial derivatives and set them equal to zero to locate critical points. The function has no explicit dependence on y other than being multiplied directly, which means y=0 is always a critical line. By examining the behavior around y=0, we can determine the nature of this critical line. Since the exponentiated variable x only grows or decreases exponentially and never reaches a maximum or minimum except at negative infinity, no local maximum or minimum exists due to the multiplier y.

Therefore, the correct answer is A. No local maximum or minimum.