Question 70
Let $f : [0, 1] \to \mathbb{R}$ be a monotone increasing differentiable function, with a continuous derivative, such that $f(0) = 0$. Let $g : [0, 1] \to \mathbb{R}$ be the function defined by
$$
g(x) = f(x) + (x-1)f'(x), \quad \text{for any } x \in [0, 1].
$$
a) Show that
$$
\int_{0}^{1} g(x) \, dx = 0.
$$
$$
g(x) = f(x) + (x-1)f'(x), \quad \text{for any } x \in [0, 1].
$$
a) Show that
$$
\int_{0}^{1} g(x) \, dx = 0.
$$