Question
Let g(x) be a continuous function such that 0ò1 g(t) dt = 2. Let f(x) = 1/2 0òx (x-t)2 g(t) dt then find f '(x) and hence evaluate f "(x).
Joshi sir comment
f(x) = 1/2 0òx (x-t)2 g(t) dt
or f(x) = 1/2 0òx x2 g(t) dt + 1/2 0òx t2 g(t) dt - 0òx x t g(t) dt
or f(x) = 1/2 x2 0òx g(t) dt + 1/2 0òx t2 g(t) dt - x 0òx t g(t) dt
now use newton leibniz formula for middle function and product rule for first and last functions for finding f' (x)
similarly get f'' (x)