# Suppose deltaf(a,b)=4i-3j. Find a unit vector so that:1.Du(f)(a,b)=02.Du(f)(a,b) is maximal3.Du(f)(a,b) is minimalWHAT DOES THIS MEAN?!

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Expert Answers

sciencesolve | Certified Educator

You need to use directional derivatives to solve the problem such that:

`D_u f = grad f*bar u`

`D_u f = |grad f|*|baru|*cos theta`

If `D_u f = 0` , then `cos theta = 0` , hence `theta = pi/2` , meaning that `grad f ` and vector `bar u` are orthogonal.

If `D_u f` is maximal, then `cos theta = 1` =>`theta = 0` , hence the vector `grad f` is upon vector `bar u` `. `

If `D_u f ` is minimal, then `cos theta = -1` =>`theta = pi` , hence the vectors `grad f ` and `bar u` have opposed directions.

**Hence, considering the conditions provided by problem yields `D_u f = 0` if `theta = pi/2, D_u f` is maximal if `theta = 0` and `D_u f` is minimal is `theta = pi` .**