Question

Find two positive numbers x and y such that their sum is 35 and the product x 2 y 5 is a maximum

Answer 1

It is given that sum of two positive numbers x and y is 35 and the product x 2 y 5 is maximum.

As sum of x and y is 35, so,

x+y=35

Then,

y=35−x

Let P( x ) be the product of the given numbers,

P( x )= x 2 y 5

Substitute y=35−x in the function,

P( x )= x 2 ( 35−x ) 5

Differentiate the above function with respect to x,

P ′ ( x )=2x ( 35−x ) 5 +5 ( 35−x ) 4 x 2 ( −1 ) =x ( 35−x ) 4 [ 2( 35−x )−5x ] =x ( 35−x ) 4 ( 70−7x ) =7x ( 35−x ) 4 ( 10−x )

Differentiate above equation with respect to x,

P ″ ( x )=7 ( 35−x ) 4 ( 10−x )+7x[ −( 35−x )−4 ( 35−x ) 3 ( 10−x ) ] =7 ( 35−x ) 3 [ ( 35−x )( 10−x )−x( 35−x )−4x( 10−x ) ] =7 ( 35−x ) 3 [ 350−45x+ x 2 −35x+ x 2 −40x+4 x 2 ] =7 ( 35−x ) 3 ( 6 x 2 −120x+350 )

Put P ′ ( x )=0,

7x ( 35−x ) 4 ( 10−x )=0 x=0,35,10

When x=0, then y=35 and the product of two number will be zero thus x=0 is not possible.

When x=35, then y=0 and the product of two number will be zero thus x=35 is also not possible.

Also, when x=10,

P ″ ( x )=7 ( 35−10 ) 3 ( 6 ( 10 ) 2 −120( 10 )+350 ) =7 ( 25 ) 3 ( −250 ) <0

This shows that the value of the function is maximum when x=10 and,

y=35−10 =25

Therefore, x=10 and y=25.