State the intermediate value theorem.(b.) Prove that there is some number x such that x2 + x − 1 = 0.

(a.) The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in the interval [a, b] such that f(c) = k.

(b.) To prove that there is some number x such that x^2 + x - 1 = 0, we can use the Intermediate Value Theorem.

Step 1: Define the function f(x) = x^2 + x - 1.

Step 2: Choose two numbers a and b such that f(a) < 0 and f(b) > 0. For example, let's choose a = -2 and b = 1. Then, f(-2) = (-2)^2 + (-2) - 1 = 4 - 2 - 1 = 1 and f(1) = (1)^2 + 1 - 1 = 1 + 1 - 1 = 1.

Step 3: Since f is a polynomial, it is continuous everywhere, including on the interval [-2, 1].

Step 4: By the Intermediate Value Theorem, since f(-2) < 0 and f(1) > 0, there must be some number c in the interval [-2, 1] such that f(c) = 0. This c is the solution to the equation x^2 + x - 1 = 0.