Let f: R R be a polynomial function of degree four having local extrema at x = -2 , x = 0 , and x = 1 . If f(0) = 0 and f(1) = 5 , then the value of _ x 0 f(x) x^2 is equal to :

Step-by-step solution

Since f(x) is a polynomial of degree four having local extrema at x = -2 , x = 0 , and x = 1 , its derivative f^ (x) is a cubic polynomial with roots at these points. Let f^ (x) = k x (x - 1) (x + 2) = k (x^3 + x^2 - 2x) . Integrating f^ (x) with respect to x , we get: f(x) = k ( x^4 4 + x^3 3 - x^2 ) + C Given f(0) = 0 , we have C = 0 . Also given f(1) = 5 : k ( 1 4 + 1 3 - 1 ) = 5 k ( 3 + 4 - 12 12 ) = 5 k ( - 5 12 ) = 5 k = -12 Substituting the value of k back into f(x) : f(x) = -12 ( x^4 4 + x^3 3 - x^2 ) = -3x