Two positive numbers x and y such that \(x+y = 60\) and \(xy^3\) is maximum are :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(x=15, y=45\)
Since x = 60 − y, the product becomes a function of y alone, with 0 < y < 60:
\(P(y) = (60 - y)\,y^3 = 60y^3 - y^4\)
Setting the first derivative equal to zero:
\(P'(y) = 180y^2 - 4y^3 = 4y^2(45 - y)\)
\(P'(y) = 0 \Rightarrow y = 45\)
Second derivative test:
\(P''(y) = 360y - 12y^2\)
\(P''(45) = 16200 - 24300 = -8100 \lt 0\)
So the product is maximum at y = 45, and x = 60 − 45 = 15.
NCERT (2026-27 edition):
Maxima and Minima
