1. Let f be differentiable on [a, b]. Suppose that f ′(x) ≥ 0 for all x ∈ [a, b] and that f ′ is not identically zero on any subinterval of [a, b]. Prove that f is strictly increasing on [a, b]. 2....

1. Let f be differentiable on [a, b]. Suppose that f ′(x) ≥ 0 for all x ∈ [a, b] and that f ′ is not identically zero on any subinterval of [a, b]. Prove that f is strictly increasing on [a, b].

2. Let f be differentiable on R. Suppose that f (0) = 0 and that 1 ≤ f ′(x) ≤ 2 for all x ≥ 0. Prove that x ≤ f (x) ≤ 2x for all x ≥ 0.

3. Suppose that f is differentiable on R and that f (0) = 0, f (1) = 2, and f (2) = 2.

(a) Show that there exists c1 ∈ (0, 1) such that f ′(c₁) = 2.

(b) Show that there exists c2 ∈ (1, 2) such that f ′(c₂
) = 0.

(c) Show that there exists c3 ∈ (0, 2) such that  f ′(c₃
) =
.