On x=y=t,f=t/2,so f is unbounded. Off zero,f_x=2xy³/(x²+y²)². Using |xy|≤(x²+y²)/2 and y²≤x²+y² gives |f_x|≤1. At the origin f_x=0 from the x-axis difference quotient. Thus **B**.
More Multivariable Calculus questions
GATE 2022 ST Q63– A point P=(x,y,z) lies on z=xy+5 and x+y+z=1 and is nearest to the origin. Find its…
GATE 2021 ST Q55– Let f(x,y)=8x²−2y. If M,m are its maximum and minimum on x²+y²=1,find M−m,to two decimals.