Q. 40 - Calculus

Question

In Exercises 31–52, find the relative maxima, relative minima, and saddle points for the given functions. Determine whether the function has an absolute maximum or absolute minimum as well. $g(x,y)=e^x\cos y$

Step-by-Step Solution

Answer

There are no maximum, minimum or saddle points for the given function. 

Step 1. Given information

A function, $g(x,y)=e^x\cos y$

Step 2. Finding the maximum, minimum and saddle points

$$\begin{gathered} \mathrm{The} \mathrm{first}-\mathrm{order} \mathrm{partial} \mathrm{derivatives} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{are}: \\ {g}_x(x,y)=\frac{\partial g}{\partial \mathrm{x}}=e^x\cos y \mathrm{and} g_y(x,y)=\frac{\partial g}{\partial \mathrm{y}}=-e^x\sin y \\ \mathrm{Now}, \mathrm{solve} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations}: e^x\cos y=0 \mathrm{and} -e^x\sin y=0, \\ \mathrm{Since}, \mathrm{there} \mathrm{exists} \mathrm{no} x \mathrm{satisfying} \mathrm{both} \mathrm{equations}, \mathrm{there} \mathrm{are} \mathrm{no} \mathrm{maximum} \\ \mathrm{or} \mathrm{minimum} \mathrm{or} \mathrm{saddle} \mathrm{points} \mathrm{and} \mathrm{hence}, \mathrm{there} \mathrm{are} \mathrm{no} \mathrm{absolute} \mathrm{maximum} \mathrm{or} \\ \mathrm{absolute} \mathrm{minimum} \mathrm{points}. \end{gathered}$$