Use the intermediate value theorem to show that f(x) = 2x³ - 3 has a zero in the interval [1, 2].

To show that f(x) = 2x^3 - 3 has a zero in the interval [1, 2] using the Intermediate Value Theorem (IVT), we need to check two conditions: Step 1: Verify that the function is continuous on the given interval. The function f(x) = 2x^3 - 3 is a polynomial function. Polynomials are continuous for all real numbers, and therefore, f(x) is continuous on the closed interval [1, 2]. Step 2: Evaluate the function at the endpoints of the interval. Calculate f(1): f(1) = 2(1)^3 - 3 = 2(1) - 3 = 2 - 3 = -1 Calculate f(2): f(2) = 2(2)^3 - 3 = 2(8) - 3 = 16 - 3 = 13 Step 3: Apply the Intermediate Value Theorem. We have f(1) = -1 and f(2) = 13. Since f(1) is negative and f(2) is positive, and f(x) is continuous on [1, 2], the Intermediate Value Theorem states that there must exist at least one value c in the open interval (1, 2) such that f(c) = 0. Since f(c) = 0 means that c is a zero of the function, we have shown that f(x) = 2x^3 - 3 has a zero in the interval [1, 2]. That's 2 down. 3 left today — send the next one.