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# Unit vector in direction of max increase of f(x,y,z)

1. ### musicmar

100
1. The problem statement, all variables and given/known data
Find the unit vector e at P=(0,0,1) pointing in the direction along which f(x,y,z)=xz+e-x2+y increases most rapidly.

3. The attempt at a solution
In order to find the direction where f increases most rapidly, I found the second derivative of f.
I don't know how to put the curly d's in here, but

<(d2f/dx2,d2f/dy2,d2f/dz2>=<4e-x2+y,e-x2+y,0>

The second derivative should be zero where f increases the most rapidly, but I'm not sure what do do with the point or how to set the second derivative equal to zero from this point.

2. ### Kevin_Axion

921
Here just click on this and copy this code:

$$\frac{\partial^2f}{\partial x^2},\frac{\partial^2f}{\partial y^2},\frac{\partial^2f}{\partial z^2}=4e^{-x^2+y},e^{-x^2+y},0$$ or you can just write $$\nabla^2 f$$

Last edited: Oct 11, 2010
3. ### musicmar

100
But that doesn't help me answer the question.

4. ### Kevin_Axion

921
I know I'm only in the 11th grade and I know very little multi-variable calculus. I was just making the question more presentable so people who have taken this course will have a better reception and hence will answer your question.

5. ### musicmar

100
Well, thanks for showing me how to enter partial derivatives, anyway.

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