1. Let f be continuous on [a, b] and differentiable on (a, b). Prove that for any x and h such that a ≤ x ∈ (0, 1) such that f (x + h) – f (x) = h f ′ (x + α h). 2. A function f is said to be...

1. Let f be continuous on [a, b] and differentiable on (a, b). Prove that for any x and h such that a ≤ x <>∈ (0, 1) such that f (x + h) – f (x) = h f ′ (x + α h).

2. A function f is said to be increasing on an interval I if x_1
2
in I implies

that f (x₁) ≤ f (x₂). [For decreasing, replace f (x₁) ≤ f (x₂) by f (x₁) ≥ f (x₂).] Suppose that f is differentiable on an interval I. Prove the following:

(a) f is increasing on I iff f ′(x) ≥ 0 for all x ∈ I.

(b) f is decreasing on I iff f ′(x) ≤ 0 for all x ∈ I