Let f(x, y, z) = x2y + z and r(t) = (e2t, 2, t2). Compute d/dt[f(r(t))] 4e2t + 2

Question

Let f(x, y, z) = x2y + z and r(t) = (e2t, 2, t2). Compute d/dt[f(r(t))] 4e2t + 21 4e2t 8e4t + 2t 4xye2t + 2t 8e2t + x2 + 2t None of these What is the maximum value of f(x, y) = 4x2 - y2 on the region where x2 + y2 1? 4 2 -1 -2 0 None of these What is the maximum value of f (x, y) = x2 - x - y on the region where {(x, y):0 x 1,0 y 1}? -2 -3/2 -5/4 -1 -3/4 None of these What point on the plane 2x - 2y + 4z = 14 is closest to the point (2, 0, -1)? (3, 0, 2) (3/2, 0, -1/2) (19/6, -7/6, 4/3) (19/6, 4/3, -1/2) (-1/2, -7/6, 0) None of these

Explanation / Answer

4) x = e^2t , y=2 , z=t^2

f(r(t)) = 2 e^4t + t^2

df(r(t)) /dt = 2 *4 e^4t + 2t

option C

5)r=Fxx ,s=Fxy, t= Fyy

if [rt-s^2] < 0 then F(x,y) is neither maximum nor minimum

Here Fx =8x ,r = Fxx =8

Fy = -2y , t =Fyy = -2

s =Fxy =0

rt-s^2 =-16 <0

F(x,y) is neither maximum nor minimum

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