Let the function f(x) = ax^2 + b x , x 0 , be strictly decreasing in (- , 0) (0, 2) and strictly increasing in (2, ) . If f(1) = 85 , then the value of f(2) is :

Step-by-step solution

Given f(x) = ax^2 + b x Differentiating with respect to x , we get: f^ (x) = 2ax - b x^2 = 2ax^3 - b x^2 The function changes its monotonicity at x = 2 , which means x = 2 is a critical point where f^ (x) = 0 . 2a(2)^3 - b = 0 16a - b = 0 b = 16a We are given that f(1) = 85 . a(1)^2 + b 1 = 85 a + b = 85 Substituting b = 16a into the above equation: a + 16a = 85 17a = 85 a = 5 So, b = 16(5) = 80 . The function is f(x) = 5x^2 + 80 x . Now, we find f(2) : f(2) = 5(2)^2 + 80 2 = 5(4) + 40 = 20 + 40 = 60 . Answer: 60