Question

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Answer 1

It is given that the sum of two numbers is 16 and sum of their cubes is minimum.

Let the first number is x, then the second number will be 16−x.

Let, S be the sum of the cubes of two numbers and since the summation of cube of both the numbers is minimum, so, S( x )= x 3 + ( 16−x ) 3 is minimum.

Differentiate the function with respect to x,

S ′ ( x )=3 x 2 −3 ( 16−x ) 2 (1)

Put S ′ ( x )=0,

3 x 2 −3 ( 16−x ) 2 =0 3 x 2 −3( 256+ x 2 −32x )=0 3 x 2 −768−3 x 2 +96x=0 x=8

Differentiate equation (1) with respect to x,

S ″ ( x )=6x+6( 16−x ) S ″ ( 8 )=6( 8 )+6( 16−8 ) =48+48 =96

This shows that the function is positive, so, x=8 is the point of minima.

As the first number is x=8, the second number will be,

y=16−x =16−8 =8

Therefore, the two positive numbers are 8 and 16.