I am working through a problem which asks for the following: $$ f(x,y,z) = x^2y + xy^2 + yz^2 $$ Calculate the gradient of this function and determine all points (u) such that $$ u \in \mathbb R^3, \nabla f(u) = 0$$
First I calculated the gradient which I believe to be $$ \nabla f = (2xy + y^2, x^2 + 2xy + z^2, 2yz)$$ The second part of the question is where I am having difficulties.
I then proceeded to set up a set of three simultaneous equations by letting each component in the gradient I have calculated to be equal to zero. I then did the algebra and solved for each variable and found that x = y = z = 0.
What I do not understand is how to use the Hessian matrix and other processes to check that the point (0,0,0) is not the only value where the gradient of the function is equal to zero. I have calculated the Hessian matrix and calculated its determinant using Matlab which I found to be $$ -8x^2y - 8xy^2 - 8yz^2$$ I am not sure how this helps but in the notes we have covered the first and second derivative tests but I do not understand how to applying these to find the other points where the gradient may also be zero.
Thank you,
Michael
