+91-8076753736 support@wiredfaculty.com
logo
logo

Sponsor Area

Application Of Derivatives

Question
CBSEENMA12035494
Wired Faculty App

Find two positive numbers a and y such that x + y = 40 and x y3 is maximum.

Solution

Here x + y = 40 ⇒ y = 40 – x ...(1)
Let f (x) = x y3 = x (40 – x)3
f'(x) = straight x straight d over dx left square bracket left parenthesis 40 minus straight x right parenthesis cubed right square bracket space plus space left parenthesis 40 minus straight x right parenthesis cubed. space straight d over dx left parenthesis straight x right parenthesis
= x [3(40 – x)2 (–1)] + (40 – x)3  
= – 3 x (40 – x)2 +(40 – x) 3
= (40 – x)2 (– 3 a + 40 – x) = (40 – x)2 (– 4 x + 40)
= 4 (40 – x)2 (10 – x)
f ' (x) = 0 ⇒  4 (40 – x)2 (10 – y) = 0 ⇒ x = 10, 40
Rejecting a = 40 as 0 < x < 40, we get, x = 10
When x < 10 slightly, f ' (x) = 4 (+) (+) = +ve
When x > 10 slightly, f ' (x) = 4 (+)(–) = –ve
∴ at x = 10, f ' (x) changes from +ve to –ve
∴ f (x) has local maximum at x = 10
But f (x) has only one extreme point x = 10
∴ f (x) is maximum when x = 10, y = 40 – 10 = 30
∴ x = 10, y = 30.

 

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation