Find the local maximum and minimum values and saddle point(s) of the function. I

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 your answers as a comma-separated list. If an answer does not exist, enter DNE.)

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) = 5y cos x, 0 x 2 pi local maximum value(s) local minimum value(s) saddle point(s) (x, y, f) =

Explanation / Answer

df(x,y)/dx = -5ysinx

df(x,y)/dy = 5cosx

So fx = -5ysinx and fy = 5cosx.

d/dx(fx) = fxx = -5ycosx

d/dy(fy) = fyy = 0

d/dy(fx) = fxy = -5sinx.

To find the critical points, fx=0 and fy=0

So, -5ysinx =0 and 5cosx=0

5cosx= 0 means x=pie/2 or 3/2*pie So sinx = 1 or -1.

Putting this value of sinx in first equation, we get y=0.

So, the critical points are (pie/2,0) and (3pie/2,0).

Second derivative test:

D = (fxx * fyy) - (fxy)^2 = (-5ycosx)*0 - (-5sinx)^2 = -25 (sinx)^2

For both the points (pie/2,0) and (3pie/2,0), (sinx)^2 = 1.

So, D=-25.

As D<0 both of these are saddle points.

Thus, (pie/2,0) and (3pie/2,0) are saddle points.

Local maximum and minimum values don't exist.

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