Find the maximum rate of change of a multivariable function
So i have this function:
$$f(x,y,z) = 4x^3yz^2+2xz^3+xyz$$
And the question is: What is the direction u for this directional derivative $f'(u,(1,-1,-1))$ to be maximal and what is the direction u for this directional derivative?
I know that $\bigtriangledown f|(1,-1,-1)=(-13,3,13)$, should i just do $\sqrt{-13^2+3^2+13^2}$?
if not, what is the correct way to do it?
Thank you
$\endgroup$ 11 Answer
$\begingroup$You are correct. The gradient at a point will give you the direction of maximum increase in the value of the function. Its direction will be $\frac{\nabla f}{|\nabla f|}$ In your case:
$$\nabla f = (12x^2yz^2+2z^3+yz,4x^3z^2+xz,8x^3yz+6xz^2+xy)$$
Therefore,
$$\nabla f(1,-1,-1) = (-13,3,13)$$
As you have already calculated.
It's direction would be $$\frac{(-13,3,13)}{\sqrt{2(13^2)+9}} \approx (-0.698,0.161,0.698)$$
$\endgroup$ 9
