In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^3−4x^2+2; between 0 and 1

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, then it must take on every value between those two points at least once. This theorem is crucial for proving the existence of roots in a given interval, as it guarantees that if the function changes signs over that interval, a real zero exists.

Polynomials

Polynomials are mathematical expressions consisting of variables raised to whole number powers and their coefficients. They are continuous and differentiable everywhere on the real number line, which makes them suitable for applying the Intermediate Value Theorem. Understanding the behavior of polynomials, such as their end behavior and turning points, is essential for analyzing their roots.

Sign Change

A sign change occurs when a function's value transitions from positive to negative or vice versa. In the context of the Intermediate Value Theorem, identifying a sign change between two points indicates that there is at least one real zero in that interval. For the polynomial f(x) = x^3 - 4x^2 + 2, evaluating the function at the endpoints of the interval [0, 1] will help determine if a sign change occurs.