Suppose fx y z x2e4yz Compute nambla f Find the maximum rat

Suppose f(x. y, z) = x^2e^4yz. Compute nambla f. Find the maximum rate of change of f at p = (2,1,4) and and find a vector in the direction of this maximum increase.

Solution

f=x²e^4yz

f_x=2xe^4yz,f_y=x²4ze^4yz,f_z=x²4ye^4yz

a)f=<2xe^4yz,x²4ze^4yz,x²4ye^4yz>

b)(x,y,z)=(2,1,4)

f=<4e¹⁶,64e¹⁶,16e¹⁶> is direction of maximum increase

maximum rate of change =|f|=e¹⁶[4²+64²+16²]

|f|=e¹⁶(4368)

|f|=66.1e¹⁶