Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.

Let the two numbers be a and b.


Given: a + b = 15


Assume, S = a2b3


S = a2(15 - a)3


= 2a(15 - a)3 - 3a2(15 - a)2


Condition maxima and minima is



⇒ 2a(15 - a)3 - 3a2(15 - a)2 = 0


⇒ a(15 - a)2 {2(15 – a) – 3a} = 0


⇒ a(15 - a)2 {30 – 5a} = 0


⇒ a = 0, 15, 6


= 2(15 - a)3 - 6a(15 - a)2 - 6a(15 - a)2+ 3a2(15 - a)


= (15 - a){2(15 - a)2 - 12a(15 - a)+ 3a2}


= (15 - a){2(15 - a)2 - 12a(15 - a)+ 3a2}


For S to minimum, > 0


a = 0 minima


a = 6 maxima


a = 15 point of inflection


Hence, two numbers are 0 and 15


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