Question

Find the local maximum and minimum values and saddle point(s) of the function. If you have three dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter NONE in any unused answer blanks.) f(x, y) = 4ex cos(y)

f(x, y) -4ex cos(y) maximum (larger x value) minimum (larger x value) saddle points ) (smallest x value) )(largest x value)

Answer 1

Answer

The given function is

f(x,y) = 4e* cos(y)

The above function is a continuous function and it is defined for all real x and y.

To find the local maximum/local minimum/saddle point, we have to set all the partial derivatives of the function to 0.

$f_x=\frac{\partial }{\partial x}\left (4e^x\cos(y) \right ) =4\cos(y)\cdot e^x =4e^x\cos(y)$

f(4e cos(y))- sin(y))-4e sin(y)

$f_x =4e^x\cos(y)=0$

f,--kr sin (y) = 0

We know that e^x is positive for all real x. Hence,

e" coS

4e* sin(y) = 0 sin(y) = 0

This system has no solution as cos(y) and sin(y) can never be zero at the same time for any y.

Since there exists no point that satisfies both f_x = 0 and f_y = 0, there is no local maximum/local minimum/saddle point.

The below graph of f(x,y) and viewpoints verifies the answer.

From the below graph, we can conclude that the given function has no local maximum/local minimum/saddle points.

5 3 2

5 4 4