Question

Proof of the Local Extreme Value Theorem Prove Theorem 4.2 for a local maximum: If \(f\) has a local maximum value at the point \(c\) and \(f^{\prime}(c)\) exists, then \(f^{\prime}(c)=0 .\) Use the following steps. a. Suppose \(f\) has a local maximum at \(c .\) What is the sign of \(f(x)-f(c)\) if \(x\) is near \(c\) and \(x>c ?\) What is the sign of \(f(x)-f(c)\) if \(x\) is near \(c\) and \(x

Short answer

Question: Prove the Local Extreme Value Theorem for a local maximum. If a function \(f\) has a local maximum value at point \(c\) and the derivative of \(f\) at \(c\) exists, show that \(f'(c) = 0\). Answer: Following the steps in the provided solution, we found the sign of \(f(x) - f(c)\) when \(x\) is near \(c\) and \(x > c\) or \(x < c\) to be non-positive. Then, we analyzed the limit of \(\frac{f(x)-f(c)}{x-c}\) as \(x \rightarrow c^{+}\) and \(x \rightarrow c^{-}\) separately, finding that \(f'(c) \leq 0\) when \(x \rightarrow c^{+}\) and \(f'(c) \geq 0\) when \(x \rightarrow c^{-}\). Combining these results, we concluded that the only possible value for \(f'(c)\) that satisfies both conditions is \(0\). Hence, \(f'(c) = 0\), proving the Local Extreme Value Theorem for a local maximum.

Step-by-step solution

a. Sign of \(f(x) - f(c)\) when \(x\) is near \(c\) and \(x > c\) or \(x < c\)

Since \(f\) has a local maximum at point \(c\), this means that the value of the function \(f(x)\) is smaller or equal to \(f(c)\) for values of \(x\) near \(c\). So, - If \(x > c\) and \(x\) is near \(c\), then \(f(x) \leq f(c)\). Therefore, the sign of \(f(x) - f(c)\) would be non-positive (i.e., \(\leq 0\)). - If \(x < c\) and \(x\) is near \(c\), then \(f(x) \leq f(c)\). Therefore, the sign of \(f(x) - f(c)\) would be non-positive (i.e., \(\leq 0\)).

b. Examine the limit of \(\frac{f(x)-f(c)}{x-c}\) as \(x \rightarrow c^{+}\)

We know that \(f'(c)\) exists and it's defined by $$f'(c) = \lim_{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$$ Let's examine this limit as \(x \rightarrow c^+\) (i.e., \(x\) approaches \(c\) from the right): $$\lim_{x \rightarrow c^{+}} \frac{f(x)-f(c)}{x-c}$$ From part a, we know that the sign of \(f(x)-f(c)\) is non-positive \((\leq 0)\) when \(x > c\). Also, \(x-c > 0\) when \(x > c\). Thus, the limit above is non-positive. Therefore, $$f'(c) \leq 0 \text{ when } x \rightarrow c^{+}$$

c. Examine the limit of \(\frac{f(x)-f(c)}{x-c}\) as \(x \rightarrow c^{-}\)

Now let's examine the limit as \(x \rightarrow c^-\) (i.e., \(x\) approaches \(c\) from the left): $$\lim_{x \rightarrow c^{-}} \frac{f(x)-f(c)}{x-c}$$ From part a, we know that the sign of \(f(x)-f(c)\) is non-positive \((\leq 0)\) when \(x < c\). Also, \(x-c < 0\) when \(x < c\). Thus, the limit above is non-negative. Therefore, $$f'(c) \geq 0 \text{ when } x \rightarrow c^{-}$$

d. Combine parts (b) and (c) to conclude that \(f'(c) = 0\)

From parts b and c, we have: $$f'(c) \leq 0 \text{ when } x \rightarrow c^{+}$$ and $$f'(c) \geq 0 \text{ when } x \rightarrow c^{-}$$ Since \(f'(c)\) exists and is non-positive when approaching \(c\) from the right and non-negative when approaching \(c\) from the left, the only possible value for \(f'(c)\) that satisfies both conditions is \(0\). Therefore, we can conclude that: $$f'(c) = 0$$ This completes the proof of the Local Extreme Value Theorem for a local maximum.

Key concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This field is a major part of modern mathematical education and has two primary areas: differential calculus and integral calculus. Differential calculus concerns the rate at which quantities change, while integral calculus is about the accumulation of quantities and the areas under and between curves.

Understanding calculus is essential for solving complex problems in physics, engineering, economics, and many other fields. It allows us to comprehend changes in systems and provides a framework to model real-world situations. The problem at hand, involving the Local Extreme Value Theorem, is a concept within differential calculus. It utilizes the idea of the derivative, which represents the rate of change of a function at a given point, to identify local maximums of functions.

Local Maximum

A local maximum of a function is a point where the function value is greater than or equal to the value at nearby points. More formally, if there is an interval around a point \(c\), such that \(f(c) \geq f(x)\) for all \(x\) in that interval, then \(f(c)\) is considered a local maximum. A crucial aspect of detecting such points involves analyzing the behavior of the function around those points.

Analyzing a function's behavior can be done by examining the sign of the difference \(f(x) - f(c)\) near the point of interest, as in our exercise. If \(f\) has a local maximum at \(c\), and \(x\) is close to but greater than \(c\), then \(f(x)\) will not exceed \(f(c)\), indicating \(f(x) - f(c)\) will be non-positive. Understanding this property is vital in determining the points where the function reaches its local highs, which are significant in various applications like optimizing functions and analyzing data trends.

First Derivative Test

The First Derivative Test is a critical tool in calculus for determining whether a function has a local maximum or minimum at a certain point. It involves examining the derivative of the function before and after the point. The test states that if the derivative changes from positive to negative at a point, the function has a local maximum there. Conversely, if the derivative changes from negative to positive, the function has a local minimum.

In the step-by-step solution, we evaluated the limits of the derivative as \(x\) approached \(c\) from both directions. If the limit from the right is less than or equal to zero, and the limit from the left is greater than or equal to zero, the First Derivative Test implies that \(f'(c)\) must be zero. This situation indicates that at point \(c\), there is neither an increase nor a decrease in the function's value, suggesting that \(c\) is a point of local extremum—in this case, a local maximum. Knowing how to apply this test is crucial for analyzing the behavior of functions and finding their extreme values.