Question

(9 points) Consider the bowl-shaped paraboloid z = f(x, y) = 4 + x2 + 3y2. (a) If you are located on the paraboloid at the point (2, - *), in which direction should you move in order to ascend on the surface at the maximum rate? What is the rate of change? (b) If you are located at the point (2, -), in which direction should you move in order to descend on the surface at the maximum rate? What is the rate of change? (c) At the point (3,1), in what direction(s) is there no change in the function values?

Answer 1

Given foy)= 4+ x² + 3y2 The partial dervative of this function are». 3f 2 of by and the gradient vector will be 04: (4:34) = (2x, 6y) 07(2,-2) - (4,-3) If you are located at (2,-)) the direction in which you will ascund on the surface at the maximum rate (4-3) The value of this rate will be 102C0-) 1: (4)2+(-3)?: 50

-) = (-4,3) (6) If you are located at (2,-1) then the direction you will desund on the surface at maximum nate us -Of(2,4)=(-4,3) Rate of charge of (2-1) = (-4)3 + 6)² = 511 To (c) there is no change in the function values if we move nadirection orthopmal to 26,67, so there two directions are parallel to 26,-67 in terms of unit vector U IC-1, 1 > and uel <1,017 V2