# Finding lines when gradient function = 0

1. Feb 28, 2014

### J_M_R

1. The problem statement, all variables and given/known data

Consider the function f(x,y) = cos(x^2+3y).

Write down the gradient of f. Then find the lines in the x-y plabe where ∇f = 0

2. Relevant equations

∇f = (∂f/∂x,∂f/∂y)

3. The attempt at a solution

-2xsin(x^2+3y) = 0

sin(x^2+3y) = 0
y = -(1/3)x^2

and

-3sin(x^2+3y) = 0

sin(x^2+3y) = 0
y = -(1/3)x^2

Should I be getting the same answer? I also think I have not understood what the question is asking as the next question goes on to say find the line where ∇f is a non zero vector pointing in the y-direction. For this I have the same answer again!

2. Feb 28, 2014

### Staff: Mentor

This is only one possible answer. There are others.

Since the question is asking for lines where ∇f = 0, it's good that you get the same answer for the x and y components of ∇f.

What is the condition for ∇f to be pointing in the y direction?

3. Feb 28, 2014

### J_M_R

I put the x component to be equal to zero, leaving me with -2xsin(x^2+3y) = 0 and then solved as before?

4. Feb 28, 2014

### J_M_R

Apologies part of my previous reply was stuck in your quote! :

The only other possible result I can think that you can get is when -2x=0 and this would give x=0. I'm not sure if I am looking for another answer in the form y=?

5. Feb 28, 2014

### Staff: Mentor

What about $\sin(x^2+3y)$? Is it really zero only for $x^2+3y = 0$?

That's the condition on $x$. What is the condition on $y$?

Remember that you have to look at both components of ∇f at the same time.