Final answer:
The function f(x,y) = 6e^x cos(y) does not have local maximum or minimum values due to the periodic nature of the cosine and sine functions. However, it has saddle points along the lines where y = (2n+1)*pi/2 and y = n*pi for any integer n.
Step-by-step explanation:
To find the local maximum, local minimum values, and saddle point(s) of the function f(x,y) = 6excos(y), we must first calculate the first partial derivatives with respect to x and y, set them equal to zero, and find the critical points. Then, we need to calculate the second partial derivatives to apply the second derivative test for functions of two variables.
The first partial derivative with respect to x is fx(x,y) = 6excos(y), and with respect to y is fy(x,y) = -6exsin(y). Setting these equal to zero, we find that critical points occur when cos(y) = 0 and sin(y) = 0, which do not happen simultaneously.
Therefore, the function does not have local maximum or minimum values, but if cos(y) is zero, x can be any real number, and when sin(y) is zero, the same applies. This indicates the presence of saddle points along the lines where y = (2n+1)π/2 and y = nπ respectively, for any integer n.
By analyzing the second partial derivatives and applying the second derivative test, we would confirm the nature of the points as saddle points if the determinant of the Hessian matrix is negative at those points. Unfortunately, without specifying x, we cannot define a single saddle point, but we do know the lines along which they occur.