Let f(x) be a linear function such that f(2) = 9 and _ r=1 ¹⁰ f(r) = 230 . If a function g(x) is defined as g(x) = ₀^ x^2 t f(t) dt , then the value of g'(2) is :

Step-by-step solution

Let f(x) = ax + b . Given f(2) = 9 2a + b = 9 Also, _ r=1 ¹⁰ f(r) = 230 _ r=1 ¹⁰ (ar + b) = a 10 11 2 + 10b = 230 55a + 10b = 230 11a + 2b = 46 Solving 4a + 2b = 18 and 11a + 2b = 46 , we get 7a = 28 a = 4 . Then b = 9 - 2(4) = 1 . So, f(x) = 4x + 1 . Now, g(x) = ₀^ x^2 t f(t) dt By Newton-Leibniz formula, g'(x) = x^2 f(x^2) d dx (x^2) = 2x^3 f(x^2) Substitute x = 2 : g'(2) = 2(2)^3 f(2^2) = 16 f(4) Since f(4) = 4(4) + 1 = 17 , g'(2) = 16 17 = 272 . Answer: 272