Final answer:
To find the unit vectors that give the direction of steepest ascent and steepest descent at a given point, calculate the gradient vector of the function at that point and normalize it. To find a vector pointing in a direction of no change, find the critical points of the function by setting its partial derivatives to zero.
Step-by-step explanation:
To find the unit vectors that give the direction of steepest ascent and steepest descent at point P(-1,2), we need to calculate the gradient vector of the function f(x,y) at that point. The gradient vector gives the direction of steepest ascent, which is the opposite direction of steepest descent.
The gradient vector can be calculated by taking the partial derivatives of f(x,y) with respect to x and y and evaluating them at the point P.
Once we have the gradient vector, we can normalize it to obtain the unit vectors that give the direction of steepest ascent and steepest descent.
To find a vector that points in a direction of no change in the function at point P, we need to find the critical points of f(x,y) by setting its partial derivatives equal to zero and solving for x and y. The vectors with those x and y values will give the direction of no change in the function at point P.