Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
The two numbers are x and y such that x + y = 60.
⇒ y = 60 − x
Let f(x) = xy3.
∴By second derivative test, x = 15 is a point of local maxima of f. Thus, function xy3 is maximum when x = 15 and y = 60 − 15 = 45.
Hence, the required numbers are 15 and 45.