U = x13x22
p1x1+p2x2 = Y
U = (Y-p2x2 / p1)3x22
for utility maximization, dU/dx2=0
3(Y-p2x2 / p1)2x22(-p2/p1) + 2(Y-p2x2 / p1)3x2
= (Y-p2x2 / p1)2x2 ( 2Y/p1 -5p2x2/p1)
=> x2 = p1Y/p2 , 0, 2Y/5p2
maxima will occur for
x2 = 2Y/5p2
=> x1 = 3Y/5p1
U = (3Y/5p1)^3 * (2Y/5p2)^2
U = (108/3125)*Y^5 / p13p22
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