Let f(x) = x^p - x^q for x [0, 1] , where p and q are positive integers such that p > q and (p, q) = 1 . If the absolute minimum value of f(x) in the interval [0, 1] is - 256 3125 , then the value of 2p + 3q is equal to

Step-by-step solution

f'(x) = p x^ p-1 - q x^ q-1 = x^ q-1 (p x^ p-q - q) . Setting f'(x) = 0 , we get the critical point x^ p-q = q p . The minimum value of f(x) occurs at this critical point: f(x) = x^q(x^ p-q - 1) . Substituting x^ p-q = q p , we get x = ( q p )^ 1 p-q . Minimum value M = ( q p )^ q p-q ( q p - 1 ) = ( q p )^ q p-q ( q-p p ) . We are given M = - 256 3125 = - 4^4 5^5 = ( 4 5 )^4 (- 1 5 ) . Comparing the two expressions, we have q-p p = - 1 5 5q - 5p = -p 5q = 4p p q = 5 4 . Since (p, q) = 1 , we get p = 5 and q = 4 .