Roots (Zeros)


Show that the functions in Exercises 19–26 have exactly one zero in the given interval.


f(x) = x⁴ + 3x + 1, [−2, −1]

Verified step by step guidance

1. First, understand that a zero of a function f(x) is a value x = c such that f(c) = 0. We need to show that the function f(x) = x⁴ + 3x + 1 has exactly one zero in the interval [-2, -1].
2. Apply the Intermediate Value Theorem, which states that if a function is continuous on a closed interval [a, b] and takes on different signs at the endpoints, then there is at least one c in (a, b) such that f(c) = 0. First, evaluate f(x) at the endpoints of the interval: f(-2) and f(-1).
3. Calculate f(-2): Substitute x = -2 into the function: f(-2) = (-2)⁴ + 3(-2) + 1. Simplify this expression to find the value of f(-2).
4. Calculate f(-1): Substitute x = -1 into the function: f(-1) = (-1)⁴ + 3(-1) + 1. Simplify this expression to find the value of f(-1).
5. Check the signs of f(-2) and f(-1). If f(-2) and f(-1) have opposite signs, then by the Intermediate Value Theorem, there is at least one zero in the interval [-2, -1]. To show that there is exactly one zero, consider the derivative f'(x) = 4x³ + 3. Analyze the sign of f'(x) over the interval to ensure that f(x) is either strictly increasing or decreasing, which would imply exactly one zero.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function, f(x), takes on different signs at two points a and b, then it must cross zero at some point between a and b. This theorem is crucial for proving the existence of a root within a specific interval, as it ensures that the function transitions from positive to negative or vice versa.

Continuity of Polynomial Functions

Polynomial functions, like f(x) = x⁴ + 3x + 1, are continuous everywhere on the real number line. This property is essential when applying the Intermediate Value Theorem, as it guarantees that there are no breaks or jumps in the function's graph, allowing us to confidently assert the existence of a zero within a given interval.

Derivative and Monotonicity

The derivative of a function provides information about its monotonicity, indicating whether the function is increasing or decreasing. By analyzing the derivative of f(x) = x⁴ + 3x + 1, we can determine if the function is strictly increasing or decreasing in the interval [−2, −1], which helps establish that there is exactly one zero by ensuring no additional turning points exist within the interval.