Question

Find the local maximum and minimum values and saddle point(s) of the function
f(x,y) = e^x cos(y)

Answer 1

@amistre64

Answer 2

I worked out fx and fy
so\[f_x=e^x\cos(y)\text{ }f_y=-e^x\sin(y)\]
So now?

Answer 3

fxx fyy - (fxy)^2 rings a bell

Answer 4

Yes, I have to determine that as well. But first I need to find the critical points, right?
I'm a little lost... hah

Answer 5

critical points are equal to 0 or undefined

Answer 6

fx is just ignoring the y direction and focusing on how it curves along x

Answer 7

Does it work like that every single time? I.e. when you look at fx=0, you ignore all the y's if there are any in the fx? Same for the fy?

Answer 8

f = e^x cos(y)

fx = e^x cos(y) fy = -e^x sin(y)

fx = 0 when e^x = -cos(y)

by ignoring the ys, i mean that y is simply some constant during a derivative

Answer 9

fxx tells us curvative

Answer 10

fxx = e^x cos(y) fyy = -e^x cos(y)

fxy = -e^x sin(y)

Answer 11

Oh haha!

so how do I get the values when I say fx and fy = 0? Im struggling with them in problems like this one

Answer 12

i think we should get it to this point, then compare the rules:

D = fxx fyy - (fxy)^2

if D>0 and fxx > 0, then min at a,b
if D>0 and fxx < 0, then max at a,b
if D<0, then saddle point
if D=0, then its still undetermined

Answer 13

Okay

Answer 14

also, there is something about a system of equations for fx fy

fx = 0
fy = 0

solve the system of equations

Answer 15

e^x cos(y) = -e^x sin(y)

cos(y) = -sin(y) ; this is a 45 degree angle in Q2 and Q4 if my minds eye is not blind

Answer 16

Okay so

e^x cos(y) = 0
-e^(x) sin(y) = 0

How to I get the values of x and y?

In fx we know that either/both of e^x or cos y makes the answer be 0.
It cannot be e^x, right?
So we are left with cos y = 0, right?

Answer 17

Oh I see!

I can say 0 = -sin(y)/cos(y) = -tan(y)
and solve for y

plug that in to any eq to get x?

Answer 18

sounds like a winner to me

Answer 19

Was my first attempt wrong?

Answer 20

i would say -1 = tan(y) is a solution, not 0=tan(y)

Answer 21

y = 3pi/4 or -pi/4

Answer 22

I didn't get that with -tan(y) = 0. I got y = 0

Answer 23

Ah I see, I made a stupid mistake haha

Answer 24

thats because its not 0 :)

e^x cos(y) = -e^x sin(y)

-cos(y) = sin(y)

-1 = tan(y)

sin(-pi/4) = -sqrt(2)/2
cos(-pi/4) = sqrt(2)/2


sin(3pi/4) = sqrt(2)/2
cos(3pi/4) = -sqrt(2)/2

Answer 25

Thanks @amistre64 ! I got the correct answer now!

Answer 26

yay!! :)