Let \( f : [0, 1] \to \mathbb{R} \) be a continuous function such that f(0) = 0 and f(1) = 1 .
Define a function \(g: [0,1] \to \mathbb{R} \) by
\(g(x) = \int_0^x f(t) \, dt - x f(x). \)
Which of the following statements is true about the function g ?
1
There exists a point \(c \in (0,1) \) such that g(c) = 0 , guaranteed by the Intermediate Value Theorem.
2
If f is continuous on [0,1], then the function g is not guaranteed to have a root in (0,1) because g(x) might not change sign.
3
The function g(x) is always continuous on [0,1] and has a root in (0,1) if f(x) is strictly increasing on [0,1].
4
The function g(x) is continuous on [0,1] and Intermediate Value Theorem is not Applicable here.