Examine each of the statements - intermediate value theorem

Question

1. Use the Intermediate Value Theorem and Rolle's Theorem to prove that the equation x⁴ + 2x² - 2 = 0 has exactly one real solution on the interval 0 < x < 1.

2. Consider the cubic function f (x) = ax³ + bx² + cx + d, where a ≠ 0.

Prove that f can have zero, one, or two critical numbers, and give an example for each.

3. Assume that f is continuous on [-1,3] and differentiable on (-1,3). Assume also that f (-1) = 5 and f(3) = -1. Examine each of the following statements. If always true, say so, and cite a relevant theorem or fact to support your answer. If false (false means "not always true") then say so, and either explain your answer or give a counter example.

(a) There exists a number c ∈ (-1, 3) such that f'(c) = 3/2.

(b) If k ∈ (-1, 5), then there exists a number c ∈ (-1,3) such that f(c) = k.

(c) If c ∈ (-1,3), f'(c) = 0 & f"(c) = 0, then f(c) is a maximum or minimum on [-1,3].