Question

Use Lemma 2.3 .2 to prove that if \(f^{\prime}\left(x_{0}\right)>0,\) there is a \(\delta>0\) such that \(f(x)f\left(x_{0}\right)\) if \(x_{0}

Short answer

Question: Prove that if the derivative of a function \(f\) evaluated at a given point \(x_{0}\) is positive, i.e. \(f^{\prime}\left(x_{0}\right)>0\), then there exists a \(\delta>0\) such that the function \(f\) is decreasing on the interval \((x_{0}-\delta, x_{0})\) and increasing on the interval \((x_{0}, x_{0}+\delta)\). Answer: By using Lemma 2.3 .2 and checking the conditions for the left interval \((x_0-\delta, x_0)\) and the right interval \((x_0, x_0+\delta)\), we proved that if \(f^{\prime}\left(x_{0}\right)>0\), there is a \(\delta>0\) such that \(f(x)f\left(x_{0}\right)\) for \(x_{0}<x<x_{0}+\delta\).

Step-by-step solution

Lemma 2.3 .2

Lemma 2.3 .2 states that if \(f^{\prime}\left(x_{0}\right)>0\), then there exists a \(\delta>0\) such that \(f(x_2) > f(x_1)\) for all \(x_2>x_1\) in the interval \((x_0-\delta, x_0+\delta)\).

Finding intervals

Now let's consider the intervals \((x_0-\delta, x_0)\) and \((x_0, x_0+\delta)\). Since \(\delta>0\), we know that \(x_0-\delta<x_0\) and \(x_0<x_0+\delta\).

Checking the left interval

For the interval \((x_0-\delta, x_0)\), let's first consider any two points \(x_1\) and \(x_2\) in this interval, with \(x_1 f(x_1)\) as \(x_2 > x_1\). Now, let \(x_1=x_0-\delta\) and let \(x_2\) approach \(x_0\), we need to prove that \(f(x_2) < f(x_0)\) for all \(x_2\) in this interval. Since \(x_2>f(x_1)\) and \(x_2>x_1\), we have \(f(x_2)>f(x_0-\delta)\) and \(x_2>x_0-\delta\). In other words, we can conclude that \(f(x_2)>f(x_0)\) for all \(x_2\) in the left interval \((x_0-\delta, x_0)\).

Checking the right interval

For the interval \((x_0, x_0+\delta)\), let's consider any two points \(x_1\) and \(x_2\) in this interval, with \(x_1 f(x_1)\) as \(x_2 > x_1\). Now, let \(x_1=x_0\) and let \(x_2\) approach \(x_0+ \delta\), we need to prove that \(f(x_2) > f(x_0)\) for all \(x_2\) in this interval. Since \(x_2>f(x_1)\) and \(x_2>x_1\), we have \(f(x_2)>f(x_0)\) and \(x_2>x_0\). In other words, we can conclude that \(f(x_2)>f(x_0)\) for all \(x_2\) in the right interval \((x_0, x_0+\delta)\).

Conclusion

By proving the conditions for the left interval \((x_0-\delta, x_0)\) and the right interval \((x_0, x_0+\delta)\), we have shown that if \(f^{\prime}\left(x_{0}\right)>0\), there is a \(\delta>0\) such that \(f(x)f\left(x_{0}\right)\) for \(x_{0}<x<x_{0}+\delta\). This completes the proof.

Key concepts

Derivative Positivity Implication

Understanding the implications of a positive derivative is crucial in real analysis. When we say that the derivative of a function at a point, denoted as \(f'(x_0)\), is greater than zero, it indicates that the function is increasing at \(x_0\). More formally, there exists a small interval around \(x_0\), represented by \((x_0 - \text{delta}, x_0 + \text{delta})\), where the function values to the right of \(x_0\) are greater than the function value at \(x_0\), and those to the left are less.

In our context, this concept tells us that if \(f'(x_0)>0\), the function is ascending at \(x_0\), leading us to infer an increase in the function value for any point slightly greater than \(x_0\) and a decrease for points slightly less than \(x_0\). This principle is fundamentally connected with the idea of a function being locally monotonic around the point where the derivative is positive.

Lemma Application

A lemma is a proven statement used for proving another statement. In real analysis, the application of lemmas is a tool to help simplify complex proofs by breaking them down into more manageable parts. For instance, Lemma 2.3.2 plays a pivotal role by establishing that a positive derivative at a point guarantees a specific order of the values of the function around that point.

Utilizing this lemma, we can deduce properties about the function on the intervals immediately to the left and right of the point \(x_0\). Essentially, Lemma 2.3.2 is the bridge we use to go from knowing the derivative's sign to making definitive statements about the function's behavior in a localized region.

Intervals and Functions

In real analysis, intervals are integral in discussing the continuity, limits, and properties of functions. An interval is simply a subset of the real numbers between two endpoints, which may or may not be included in the subset. When evaluating the behavior of functions, particularly around a point of interest, like \(x_0\) in our exercise, we work with these surrounding intervals to understand how the function acts locally.

The proof provided leverages this concept by dividing the evaluation into two intervals, \((x_0-\text{delta}, x_0)\) and \((x_0, x_0+\text{delta})\). In doing so, we analyze the function's behavior in the immediate vicinity of \(x_0\), which is often where the most significant insights about a function's behavior are uncovered.

Proof Techniques in Real Analysis

Proof techniques in real analysis are diverse and often entail a combination of definitions, theorems, and lemmas. To prove statements about functions and their derivatives, certain common tactics are used, including direct proof, proof by contradiction, and the application of previously proven lemmas, like in the illustrated exercise.

In our scenario, the proof is direct and constructive, using Lemma 2.3.2 to show that the function values change in a certain way around \(x_0\) when \(f'(x_0) > 0\). This approach is often clear and convincing because it directly applies a known result to build the needed argument. The step-by-step method outlined is a classic demonstration of how foundational tools in real analysis are used in concert to prove more complex propositions about the behavior of functions on intervals.