Answer to Question #206747 in Calculus for Joy

Question

The product of two numbers is 4 square root of 3. Find the numbers so that the
sum 𝑆 S of the square of one and the cube of the other is as small as
possible.

Answer 1

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)