Finding lines when gradient function = 0

[J_M_R]

Homework Statement

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

Homework Equations

∇f = (∂f/∂x,∂f/∂y)

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!

 

[DrClaude]

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

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

This is only one possible answer. There are others.

Should I be getting the same answer?

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.

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!

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

 

[J_M_R]

This is only one possible answer. There are others.

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=?


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

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

 

[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=?

 

[DrClaude]

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=?

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

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

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.