Question

f(x) : [0, 5] → R, F(x) = ∫x² g(x), x ∈ [0, x], f(1) = 3

g(x) = ∫f(t) dt, t ∈ [1, x], then correct choice is

(1) F(x) has local minimum at x = 1

(2) F(x) has local maximum at x = 1

(3) F(x) has point of inflection at x = 1

(4) F(x) has no critical point

Answer 1

Answer is (1)

⇒ F'(x) = x²g(x)

⇒ F'(1) = 1.g(1) = 0 .....(1) (∴ g(1) = 0)

Now F''(x) = 2xg(x) + x²g'(x)

⇒ F''(x) = 2xg(x) + x²f(x)

⇒ F''(1) = 0 + 1 x 3 (∴ g'(x) = f(x))

⇒ F''(1) = 3  .....(2)

From (1) and (2) F(x) has local minimum at x = 1