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

1. Oct 11, 2010

### musicmar

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. Oct 11, 2010

### Kevin_Axion

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. Oct 11, 2010

### musicmar

But that doesn't help me answer the question.

4. Oct 11, 2010

### Kevin_Axion

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. Oct 11, 2010

### musicmar

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