CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
Question

Find two positive numbers x and y such that x + y = 60 and xy 3 is maximum.

Open in App
Solution

The two numbers are given such that,

x+y=60

It can be written as,

y=60x

As x y 3 is given to be maximum, so substitute the value of y in the function,

f( x )=x ( 60x ) 3

Differentiate the function with respect to x,

f ( x )= d[ x ( 60x ) 3 ] dx = ( 60x ) 3 ×1+x×3 ( 60x ) 2 ×( 1 ) = ( 60x ) 2 ( 60x3x ) = ( 60x ) 2 ( 604x ) (1)

This gives x=60 or x=15.

Differentiate equation (1) with respect to x,

f ( x )=2( 60x )( 604x )4 ( 60x ) 2 =2( 60x )[ 604x+2( 60x ) ] =2( 60x )( 1806x ) =12( 60x )( 30x )

When x=60,

f ( x )=0

When x=15,

f ( x )=12( 6015 )( 3015 ) =12( 45 )( 15 ) <0

This shows that the point of local maxima is x=15, so,

y=60x =6015 =45

Therefore, x=15 and y=45.


flag
Suggest Corrections
thumbs-up
0
BNAT
mid-banner-image