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

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different values at the endpoints, then it must take on every value between f(a) and f(b) at least once. This theorem is crucial for proving the existence of real zeros in polynomials, as it guarantees that if the function changes sign over an interval, there is at least one root within that interval.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, the polynomial f(x) = 2x^4 - 4x^2 + 1 is a continuous function, which is essential for applying the Intermediate Value Theorem. Understanding the behavior of polynomial functions, including their continuity and end behavior, is key to 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), evaluating the function at the endpoints -1 and 0 will reveal whether a sign change exists, thus confirming the presence of a real zero.