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Finding lines when gradient function = 0

  1. Feb 28, 2014 #1
    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


    -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. jcsd
  3. Feb 28, 2014 #2


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    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?
  4. Feb 28, 2014 #3
    I put the x component to be equal to zero, leaving me with -2xsin(x^2+3y) = 0 and then solved as before?
  5. Feb 28, 2014 #4
    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=?
  6. Feb 28, 2014 #5


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    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.
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