For the function fxy x2y Duf23 is zero Please show steps Ke

Solution

Given f(x,y) = x^2y

Partial derivative wrt x, 2y*(x^(2y-1))

Partial derivative wrt y, (x^2y)*log(x) here is log is to base \"e\"

Grad f (2,3) = < 2*3*(2^(2*3-1)) , 2^(2*3)* log(2) >

= < 192, 19.2659 >

Given Duf(2,3) = 0

=> < 192, 19.2659 > . U = 0 U being the unit vector

=> U is perpendicular to < 192, 19.2659 >

Direction cosines of < 192, 19.2659 > are 0.995 i + 0.099 j or approximately 0.995 i + 0.1 j

Let U = a i + b j (unit vector)

=> a^2 + b^2 =1 & 0.995a + 0.1b = 0

b = -9.95 a

a^2 + (-9.95a)^2 = 1 => a = 0.1 b =-0.995

This is the method to follow. I dunno, may have made an arithmetic mistake somewhere. Otherwise its correct