Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x^3-37x^2+50x+60 between 2 and 3

Verified step by step guidance

1. Identify the polynomial function: \( f(x) = 4x^3 - 37x^2 + 50x + 60 \).
2. Use the Intermediate Value Theorem, which states that if a continuous function changes sign over an interval \([a, b]\), then there is at least one real zero in that interval.
3. Evaluate \( f(x) \) at the endpoints of the interval: calculate \( f(2) \) and \( f(3) \).
4. Check the signs of \( f(2) \) and \( f(3) \). If \( f(2) \) and \( f(3) \) have opposite signs, then there is at least one real zero between 2 and 3.
5. Conclude that there is a real zero between 2 and 3 based on the sign change, as guaranteed by the Intermediate Value Theorem.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer. Understanding the behavior of polynomial functions is crucial for analyzing their roots or zeros.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, it must also take on any value between those two points at some point in the interval. This theorem is essential for proving the existence of real zeros in polynomial functions, as it allows us to conclude that if the function changes signs between two values, there is at least one zero in that interval.

Finding Real Zeros

Finding real zeros of a polynomial function involves determining the values of x for which f(x) = 0. This can be done through various methods, including factoring, using the Rational Root Theorem, or numerical methods like the Newton-Raphson method. In the context of the given polynomial, evaluating the function at specific points helps identify intervals where the function changes sign, indicating the presence of real zeros.