Answer in Calculus for Joy #206747
The product of two numbers is 4 square root of 3. Find the numbers so that the sum S of the square of one and the cube of the other is as small as possible.
Given ab = 4*3(1/2) , a = 4*3(1/2)/b
Given f(a,b) = a2 + b3 = (48/b2)+b3
Differentiating both sides with respect to b
f'(a,b) = 3b2 – (96/b3)
For critical points, f'(a,b) = 0
3b2 = 96/b3
b5 = 32
b = 2
Again Differentiating both sides with respect to b
f”(a,b) = 6b + 384/b4 > 0
At b = 2, f(a,b) has local minimum.
So, if b = 2, a = 2*3(1/2)