Science:Math Exam Resources/Courses/MATH105/April 2011/Question 08 (c)/Solution 1

In part (a) we computed the partial derivatives, allowing us to write the gradient vector

$\displaystyle \nabla f(x,y)=(120x^{-1/4}y^{1/4},40x^{3/4}y^{-3/4})$

hence (as calculated previously)

$\displaystyle \nabla f(81,16)=(80,135)$

Recall that a perpendicular vector to (a,b) with the same length is is (b,-a) and (-b,a). Therefore, the perpendicular vectors (of the same length as the gradient itself) are the vectors

$\displaystyle (-135,80)\quad {\text{and}}\quad (135,-80)$