Question

Proving Theorem 2 Assume that the function \(f\) has a local maximum value at the interior point \(c\) of its domain and that \(f^{\prime}(c)\) exists. (a) Show that there is an open interval containing \(c\) such that \(f(x)-f(c) \leq 0\) for all \(x\) in the open interval. (b) Writing to Learn Now explain why we may say $$\lim _{x \rightarrow c^{+}} \frac{f(x)-f(c)}{x-c} \leq 0$$ (c) Writing to Learn Now explain why we may say $$\lim _{x \rightarrow c^{-}} \frac{f(x)-f(c)}{x-c} \geq 0$$ (d) Writing to Learn Explain how parts (b) and (c) allow us to conclude \(f^{\prime}(c)=0 .\) (e) Writing to Learn Give a similar argument if \(f\) has a local minimum value at an interior point.

Short answer

Given a function \(f\) with a local maximum at an interior point \(c\), it can be proved that \(f'(c) = 0\). Similarly, for a function with a local minimum at an interior point, \(f'(c) = 0\) indicates the presence of the local minimum.

Step-by-step solution

Prove Local Maximum Condition

To prove (a), since \(f\) has local maximum value at \(c\), there exists an open interval \(I\) containing \(c\) such that for any \(x\) in \(I\), \(f(x) \leq f(c)\). Therefore, \(f(x) - f(c) \leq 0\) for all \(x\) in \(I\).

Establish Right-Hand Limit

For (b), since \(x\) approaches \(c\) from the right side (i.e., \(x > c\)), the denominator \(x - c\) in the limit expression is greater than 0. Given that \(f(x) - f(c) \leq 0\) from the previous step, the entire fraction \(\frac{f(x)-f(c)}{x-c}\) is thus less than or equal to 0, leading to \(\lim _{x \rightarrow c^{+}} \frac{f(x)-f(c)}{x-c} \leq 0\).

Establish Left-Hand Limit

Similarly for (c), on approaching \(c\) from the left (i.e., \(x < c\)), \(x - c\) in the denominator is less than 0. Given that \(f(x) - f(c) \leq 0\), the entire fraction \(\frac{f(x)-f(c)}{x-c}\) is hence greater than or equal to 0, leading to \(\lim _{x \rightarrow c^{-}} \frac{f(x)-f(c)}{x-c} \geq 0\).

Finding the Derivative at the Point

For (d), by the definition of the derivative, \(f^{\prime}(c) = \lim_{x \rightarrow c} \frac{f(x) - f(c)}{x - c}\). Considering the results from (b) and (c), since both left and right hand limits are negative and therefore equal to 0, it can be concluded that \(f^{\prime}(c)=0\).

Argument for Local Minimum

For (e), if \(f\) has a local minimum value at an interior point rather than a maximum, the proof works similarly. The difference being that \(f(x) - f(c) \geq 0\) in the open interval \(I\) and thus \(\lim_{x \rightarrow c^{+}} \frac{f(x) - f(c)}{x - c} \geq 0\) and \(\lim_{x \rightarrow c^{-}} \frac{f(x) - f(c)}{x - c} \leq 0\), again implying \(f'_{\prime}(c) = 0\)

Key concepts

Local Maximum and Minimum

When we talk about local maximum and minimum in calculus, we're looking at specific points on a graph. These are not necessarily the highest or lowest values of the entire function, but rather the peaks and valleys within a specific region.

More formally, a function has a local maximum at a point if the function's value at that point is greater than the value at surrounding points. The reverse is true for a local minimum; the function's value is the lowest compared to its neighbors. This concept is crucial, as it helps us understand where a function reaches its most extreme values locally, and studying these points can provide insight into the behavior of the function as a whole.

Importantly, when applying the concept to exercises like the one provided, having a local extremum at point 'c' suggests that the function values around 'c' are either all higher (for a minimum) or all lower (for a maximum) than the value at 'c', within an open interval surrounding 'c'. This characteristic underpins the solutions to the given problem.

Definition of a Derivative

The definition of a derivative is central to calculus. It provides a way to measure the rate at which a function is changing at any given point. In simple terms, a derivative represents the slope of the tangent line to the function's graph at a specific point.

Formally, the derivative of a function at a point 'c' is defined by the limit \(f'_{\prime}(c) = \lim_{x \rightarrow c} \frac{f(x) - f(c)}{x - c}\), assuming this limit exists. This expression calculates the slope of the secant lines approaching the tangent line as 'x' approaches 'c'. In essence, the concept of the derivative allows us to quantify the concept of an instantaneous rate of change. When solving calculus problems, as seen in our exercise, knowing that the derivative at a local extremum should be zero helps us to set up the necessary conditions for our proofs.

Limits and Continuity

The concepts of limits and continuity are foundational to understanding calculus. A limit represents the value that a function approaches as its input approaches some point. It's like trying to find out what value y is getting closer to as x gets closer to a certain number.

Continuity, on the other hand, means that a function is unbroken or smooth; there are no jumps, holes, or gaps as you trace along the graph of the function. For a function to be continuous at a point 'c', the function must be defined at 'c', the limit as 'x' approaches 'c' must exist, and the limit of the function as 'x' approaches 'c' must equal the function's value at 'c'. In our exercise, to solve parts (b) and (c), we depend on understanding the limit of a difference quotient as 'x' approaches 'c' from either side. This concept is tied to both the definition of a derivative and ensuring that the function behaves as expected near a point of extremum.

Derivative Tests for Extrema

Derivative tests provide a practical way to locate the local maxima and minima of a function. These tests use the derivative of a function as a tool to determine where the function's rate of change switches direction.

First Derivative Test

If the derivative changes from positive to negative, we have a local maximum. Conversely, if it changes from negative to positive, it indicates a local minimum.

Second Derivative Test

Another approach is to look at the second derivative of a function. If the second derivative is greater than zero, the function is concave up, suggesting a local minimum. If it's less than zero (concave down), it implies a local maximum.

Our exercise showcases a fundamental application of the derivative test, dealing with the case where a derivative equals zero indicating potential extrema. By understanding that the derivative at a local maximum or minimum must equal zero, we can infer critical points of the function that warrant further investigation using these derivative tests.