Question

find the extreme values and saddle points of f:
f(x,y)=http://www.imagebam.com/image/268ae8405421087

Answer 1

\[f(x,y) = e^x \sin(y)\]

1) calculate the partials ...

Answer 2

fx=e^x * sin(y)
fy=e^x*cos(y)
fxx=e^x * sin(y)
fyy=e^x * -sin(y)
fxy=e^x

Answer 3

then what?

Answer 4

try finding stationary points,
\[f_x = f_y = 0\]

Answer 5

if you can find any, they are the (local) extrema and/or saddle points you seek...

Answer 6

show me how..
thanks

Answer 7

there should not be any.

\[f_x=e^x \sin(y)\]

this is just constant* e^x as y is constant in the x direction.

think about the geometry, too

if you walk in the positive x direction starting at any y value, you are always walking up a hill that follows the e^x shape.

if you walk in the positive y direction starting at any x value, you are always walking up and down a sinusoid but you always have a down slope on yr left and an up slope on yr right.

Answer 8

did this make sense?

Answer 9

actually i can't get it till now

Answer 10

in 2-D, x-y space, dy/dx = 0 is all you need to find a stationary point.

in 3-D , x-y-z, where z = f(x,y), you need to find ∂z/∂x = 0 AND ∂z/∂y = 0

in this example: z = f(x,y)=e^x sin(y)

∂z/∂x is never zero, it is always increasing as you move along the x axis.....

you will get peaks and troughs as you move along the y axis as ∂z/∂y=e^x*cos(y) and that will be zero when y = pi/2 + n pi. but the overall surface is always increasing in the x direction.

at no point will you find ∂z/∂x = 0 AND ∂z/∂y = 0