Let f : R → R be a polynomial function of degree four having extreme values at x = 4 and x = 5. If limx→0 f(x)/x² = 5, then f(2) is equal to:
Question Let f : R → R f : \mathbb{R} \to \mathbb{R} be a polynomial function of degree 4 having extreme values at x = 4 x = 4 and x = 5 x = 5 . If lim x → 0 f ( x ) x 2 = 5 , then find f ( 2 ) f(2) . Question Image Short Solution Express derivative using extreme points: Since f ( x ) f(x) is degree 4 and has extreme values at x = 4 , 5 x=4,5 , the derivative f ′ ( x ) f'(x) must vanish there: f ′ ( x ) = k ( x − 4 ) ( x − 5 ) ( x − α ) for some real α \alpha and constant k k . Degree of f ′ ( x ) f'(x) is 3, consistent with degree 4 for f ( x ) f(x) . Integrate to get f ( x ) f(x) : Let’s assume a convenient factorization for simplicity: f ′ ( x ) = A ( x − 4 ) ( x − 5 ) ( x − r ) Integrate to get f ( x ) = ∫ f ′ ( x ) d x = quartic in x + C. Use the limit condition: lim x → 0 f ( x ) x 2 = 5 ⟹ f ( x ) ∼ 5 x 2 as x → 0 So the constant term in f ( x ) f(x) is 0, and the coefficient...