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

• musicmar
In summary, the problem is asking to find the unit vector e at point P=(0,0,1) in the direction where the function f(x,y,z)=xz+e-x^2+y increases most rapidly. To do this, the second derivative of f was found to be <(d^2f/dx^2,d^2f/dy^2,d^2f/dz^2)=<4e^-x^2+y,e^-x^2+y,0>. The second derivative should be set equal to zero to find the direction of maximum increase, but the method for doing so is unclear.
musicmar

## Homework Statement

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.

## 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.

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

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.

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

## 1. What is a unit vector in the direction of maximum increase of a function?

A unit vector in the direction of maximum increase of a function is a vector with a magnitude of 1 that points in the direction of the steepest increase of the function at a given point. It represents the direction in which the function increases the most rapidly.

## 2. How is the unit vector in the direction of maximum increase calculated?

The unit vector in the direction of maximum increase can be calculated by taking the gradient of the function at a given point and normalizing it to a magnitude of 1. The gradient is a vector that points in the direction of maximum increase and has a magnitude equal to the rate of change of the function in that direction.

## 3. What is the significance of the unit vector in the direction of maximum increase?

The unit vector in the direction of maximum increase is significant because it represents the direction of greatest change in the function. It can be used to determine the direction in which the function increases the most rapidly, which can be useful in optimization problems or understanding the behavior of a function.

## 4. Can the unit vector in the direction of maximum increase change at different points on a function?

Yes, the unit vector in the direction of maximum increase can change at different points on a function. This is because the direction of maximum increase is dependent on the slope of the function at a given point, which can vary as the function changes. Therefore, the unit vector can change as well.

## 5. How is the unit vector in the direction of maximum increase used in practical applications?

The unit vector in the direction of maximum increase is used in various practical applications, such as gradient descent algorithms in machine learning, optimization problems in engineering and physics, and understanding the behavior of functions in mathematical models. It can also be used to determine the direction of steepest ascent for a given function.

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