"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero

Verified step by step guidance

1. Identify the function given in the exercise. Let's denote it as f(x).
2. Determine the domain of the function f(x) to ensure it is defined for all x in the interval of interest.
3. Use the Intermediate Value Theorem (IVT) to show that there is at least one zero. The IVT states that if a function is continuous on a closed interval [a, b] and f(a) and f(b) have opposite signs, then there is at least one c in (a, b) such that f(c) = 0.
4. To show that there is exactly one zero, demonstrate that the function is either strictly increasing or strictly decreasing on its domain. This can be done by finding the derivative f'(x) and showing that it does not change sign.
5. Conclude that since the function is continuous and either strictly increasing or decreasing, it can cross the x-axis at most once, confirming that there is exactly one zero.

Key Concepts

Roots of a Function

The roots or zeros of a function are the values of the variable that make the function equal to zero. Finding the roots involves solving the equation f(x) = 0. This concept is crucial for understanding where the graph of the function intersects the x-axis, indicating the points where the function has no value.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function, f(x), takes on different signs at two points, then it must cross zero at some point between them. This theorem is essential for proving the existence of a zero within an interval, especially when the function changes from positive to negative or vice versa.

Uniqueness of Zeros

To show that a function has exactly one zero, it is important to demonstrate that the function is either strictly increasing or decreasing, ensuring no other zeros exist. This involves analyzing the derivative of the function to confirm its monotonic behavior, which guarantees that the function crosses the x-axis only once.