Question

\(3 - 14\)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.

\(f(x,y) = {e^x}\cos y\)

Short answer

No critical, maximum, minimum or saddle point.

Step-by-step solution

Second derivative test

Suppose the second partial derivatives of\(f\)are continuous on a disk with center\({\bf{(a,b)}}\), and suppose that\({f_x}(a,b) = 0\)and\({f_y}(a,b) = 0\). Let

\(D = D(a,b) = {f_{xx}}(a,b){f_{yy}}(a,b) - {\left( {{f_{xy}}(a,b)} \right)^2}\)

(a) If\(D > 0\)and\({f_{xx}}(a,b) > 0\), then\(f(a,b)\)is a local minimum.

(b) If\(D > 0\)and\({f_{xx}}(a,b) < 0\), then\(f(a,b)\)is a local maximum.

(c) If\(D < 0\), then\(f(a,b)\)is not a local maximum or minimum.a

Find partial derivative

The given function is\(f(x,y) = {e^x}\cos y\).

\({f_x}(x,y) = {e^x}\cos y\)

And\({f_y}(x,y) = - {e^x}\sin y\)

Find critical points

Solve\({f_x} = 0\)and\({f_y} = 0\).

\({e^x}\cos y = 0\)and\( - {e^x}\sin y = 0\)

\( \Rightarrow {e^x} \ne 0,{e^x}\)never equal to zero.

No value of\(x\)exists for\({e^x} = 0\).

No value of\(y\)exists for\(\cos y = 0\) and\(\sin y = 0\).

So, no values for\(x\)and \(y\)that satisfies the equations, \({e^x}\cos y = 0\)and\( - {e^x}\sin y = 0\)Therefore, there is no critical points and there are no maximum value, minimum value or saddle points.