Question

. f is differentiable in (0, 1) and continuous on [0, 1].
(a) Show that
|f(b)| − |f(a)| ≤
Z 1
0
|f
′
(x)|dx ∀a, b ∈ (0, 1)
Hint: Start with the interval [a, b], and apply the first fundamental
theorem of calculus to it.
(b) By choosing a appropriately, show that
|f(b)| ≤
Z 1
0
|f'(x)| + |f(x)|dx ∀b ∈ (0, 1)