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

Answers (1)

let x an d y are positive two numbers 
It is given that 
x + y = 16 , y = 16 - x 
and  x^3 + y^3 is minimum
f(x) = x^3 + (16-x)^3
Now, 
f^{'}(x) = 3x^2 + 3(16-x)^2(-1)
f^{'}(x) = 0\\ 3x^2 - 3(16-x)^2 =0\\ 3x^2-3(256+x^2-32x) = 0\\ 3x^2 -3x^2+96x-768= 0\\ 96x = 768\\ x = 8\\
Hence, x = 8 is the only critical point 
Now,
f^{''}(x) = 6x - 6(16-x)(-1) = 6x + 96 - 6x = 96\\ f^{''}(x) = 96
f^{''}(8) = 96 > 0
Hence, x = 8 is the point of minima
 y = 16 - x
    = 16 - 8 = 8
Hence, values of x and y are 8  and 8 respectively 

Preparation Products

JEE Main Rank Booster 2021

This course will help student to be better prepared and study in the right direction for JEE Main..

₹ 13999/- ₹ 9999/-
Buy Now
Rank Booster NEET 2021

This course will help student to be better prepared and study in the right direction for NEET..

₹ 13999/- ₹ 9999/-
Buy Now
Knockout JEE Main April 2021 (Easy Installments)

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 4999/-
Buy Now
Knockout NEET May 2021

An exhaustive E-learning program for the complete preparation of NEET..

₹ 22999/- ₹ 14999/-
Buy Now
Knockout NEET May 2022

An exhaustive E-learning program for the complete preparation of NEET..

₹ 34999/- ₹ 24999/-
Buy Now
Exams
Articles
Questions