Question

Proof Prove that if \(f\) is continuous and has no zeros on \([a, b],\) then either $$ f(x) > 0 \text { for all } x \text { in }[a, b] \quad \text { or } \quad f(x) < 0 \text { for all } x \text { in }[a, b] $$

Short answer

Based on continuity and the Intermediate Value Theorem, \( f \) can only continuously produce positive values or continuously produce negative values in the interval \([a, b]\), as there are no zeros in this interval.

Step-by-step solution

Define the parameters

Let's start by defining the continuous function \( f \) and the interval \([a, b]\) where \( f \) has no zeros. This means that there are no points in this interval where \( f(x) = 0 \).

Apply the Intermediate Value Theorem

We know that a continuous function satisfies the Intermediate Value Theorem. According to this theorem, if a function \( f \) is continuous in the closed interval \([a, b]\) and \( N \) is any value between \( f(a) \) and \( f(b) \), then there exists at least one value \( c \) in the interval \([a, b]\) such that \( f(c) = N \). Applying this to our function, we can say that if there was a point within \([a, b]\) where \( f(x) > 0 \) and a point where \( f(x) < 0 \), given the continuity of the function, there should be a point \( c \) where \( f(c) = 0 \), which clearly contradicts our initial condition that \( f \) has no zeros in this interval.

Conclude result

Due to these arguments, our function \( f \) must always be positive (i.e., \( f(x) > 0 \) for all \( x \) in \([a, b]\)) or always be negative (i.e., \( f(x) < 0 \) for all \( x \) in \([a, b]\)). Therefore it is proved that if \( f \) is continuous and has no zeros on \([a, b]\), then either \( f(x) > 0 \) for all \( x \) in \([a, b]\) or \( f(x) < 0 \) for all \( x \) in \([a, b]\).

Key concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that describes the behavior of continuous functions on intervals. Imagine you are on a hike, starting at the base of a hill and walking up to the summit. If you started at sea level and the summit is 1,000 feet above, you must have passed through every elevation between 0 and 1,000 feet on your way to the top. This is what the IVT tells us: if a function is continuous on a closed interval, then for every value between the function's values at the endpoints, the function must take that value at some point within the interval.

In layperson terms, suppose your function represents elevation along the path, and on your journey, you have recorded the elevation at the start and end points. The IVT guarantees you that if your elevation reading continuously changes without any jumps, you will have reached every height between the start and end point elevations at least once.

Zeroes of Functions

The 'zeroes of functions' or 'roots' refer to the points where a function equals zero. They are essential because they often represent the solution to an equation or the x-intercepts on a graph. Finding zeroes is like finding where a ball tossed into the air comes back down to your hand, assuming the ground is your baseline.

Particularly, when we're dealing with continuous functions, if the function crosses the x-axis (indicating a change in the sign from positive to negative, or vice versa), then we can say there is at least one zero between those points. Here's a fact to remember: a continuous function that doesn't touch the x-axis in an interval doesn't have any zeros in that interval. It's a bit like knowing that if the ball stayed always in the air or always in your hand throughout a certain time period, then it never hit the ground during that time.

Proofs in Calculus

Proofs in calculus often come across as intimidating, but they are simply a way of using logic and rules to show why certain mathematical statements must be true. When you prove something in calculus, you're like a detective who gathers evidence and connects the dots to make a solid case.

A proof often starts with what we know to be true based on definitions, theorems, and postulates—our mathematical 'clues'. Then we follow the breadcrumbs, applying these concepts step by step, to arrive at the conclusion that must be true if we've used the clues correctly. In the case of continuous functions and zeroes, we use the 'Intermediate Value Theorem', a reliable witness, to testify that under specific circumstances, a continuous function either always stays above or below the baseline—just as our detective would conclude a suspect's whereabouts based on solid alibis.

Interval Continuity

Interval continuity refers to a function being continuous at every point within a specified interval. Think of it as a non-stop flight from one city to another without any layovers—it's smooth sailing all the way through.

Continuity on an interval is crucial for many theorems in calculus, including the IVT. It implies that the function doesn't have any holes, breaks, or jumps within the interval. For instance, if you were drawing a line without lifting your pencil off the paper, that line would be continuous. If your pencil stays on the page from point 'A' to point 'B', you've just depicted interval continuity. It's this unbroken nature of functions in an interval that allows mathematicians to draw conclusions about their behavior within that range, such as their zeroes or positive and negative values.