Question

The total number of local maxima and local minima of the function $$ f(x)= \begin{cases}(2+x)^{3}, & -3

Short answer

1

Step-by-step solution

Understand the Function

The function provided is a piecewise function with two expressions. For the first interval \( -3<x \leq -1 \) , the function is given by \( f(x) = (2+x)^3 \). For the second interval \( -1<x<2 \) , it is given by \( f(x) = x^{2/3} \). We need to examine both pieces individually to find any local maxima or minima.

Find the Critical Points for the First Interval

For the interval \( -3<x \leq -1 \) , differentiate the function \( f(x) = (2+x)^3 \) to find the critical points. \[ f'(x) = 3(2+x)^2 \] This derivative does not equal zero nor does it become undefined in the interval \( -3<x \leq -1 \), hence there are no critical points in this interval.

Find the Critical Points for the Second Interval

For the interval \( -1<x<2 \) , differentiate the function \( f(x) = x^{2/3} \) to find the critical points. \[ f'(x) = \frac{2}{3}x^{-1/3} \] Since \( f'(x) \) does not exist at \( x=0 \) (division by zero), we have one critical point at \( x=0 \) in this interval.

Determine Local Maxima or Minima at the Critical Point

The critical point \( x=0 \) needs to be tested to determine if it is a local maximum or minimum. Since the power \( 2/3 \) is a rational number and \( x \) is raised to this power, the function \( x^{2/3} \) is increasing on its interval \( -1<x<2 \) and thus \( x=0 \) is not a local extremum.

Analyze the Behavior at the Endpoints

Although local extrema are normally found at critical points, we must check the endpoints of the intervals for any changes in behavior that may imply a local maximum or minimum. For \( x=-1 \), which is an endpoint that belongs to both intervals, we examine the left-hand limit of the first piece \( (2+x)^3 \) and right-hand limit of the second piece \( x^{2/3} \). Both approach the same value \( (-1)^{2/3} = 1 \) smoothly, indicating no local extremum at the shared endpoint. There's no need to evaluate at \( x=-3 \) or \( x=2 \) since they are not within the open intervals, and local extrema cannot occur at endpoints that are not interior points of the domain of the function.

Key concepts

Piecewise Functions

Piecewise functions, like the one provided in the exercise, are defined by different expressions over different intervals of the independent variable, often representing complex or disjoint phenomena. When analyzing such functions for local extrema, it's necessary to consider each piece individually. This approach helps locate any distinctive points where the function might change its course, hence revealing potential maxima or minima. The provided function, structured with two intervals, poses a unique challenge in finding local extrema as we need to explore the continuity and derivative of each function piece separately.

In the context of our exercise, we pay close attention to the boundaries at \( x = -1 \) to ensure a precise understanding of the function’s behavior at that point, where the two pieces meet. Such attention to detail assures that students can confidently navigate the complexities of piecewise functions.

Critical Points

Critical points are the 'X marks the spot' for mathematicians seeking treasure in the form of local maxima and minima. They occur where the first derivative of a function is either zero or undefined. Understanding how to find critical points is essential because they can be potential candidates for where a function reaches its highest or lowest local values. However, it's important to note not all critical points yield a maximum or minimum, some may be inflection points where the curve changes concavity.

For our exercise, identifying the lack of critical points for the first interval and the presence of one for the second interval at \( x = 0 \) is a crucial step in the process. It highlights examining both where the derivative equals zero and where it does not exist—a fundamental approach often overlooked by students navigating function analysis.

Differentiation

Differentiation is a powerful mathematical tool, allowing us to find the rates at which quantities change, commonly referred to as derivatives. By differentiating a function, we obtain insights into its slope at any given point, enabling us to predict increasing or decreasing behavior, and more importantly for our cause, the potential places where it might not change at all—stationary points which could indicate a local extremum.

In the exercise at hand, differentiating the two parts of the function is the key to unlocking the whereabouts of critical points. It demonstrates that while differentiation is a straightforward computation, interpreting its results—such as the critical point found at \( x = 0 \)—informs us about the intricacies involved in function analysis. Here, the derivative helps us conclude that this particular point doesn't actually correspond to a maximum or minimum, an outcome sometimes counterintuitive to students.

Function Analysis

Function analysis encapsulates all the processes of deeply understanding the properties and behaviors of a function—its continuity, limits, derivatives, and the consequent identification of extrema. It's like being a detective, deducing from the various clues (properties) how a function behaves globally and locally across its domain.

Through analysis, we find that our exercise presents no change in behavior at the endpoints worth considering for local extrema. This comprehensive approach reminds students to look beyond critical points, considering all possible scenarios where a function may exhibit local maximum or minimum behaviors, including the often-neglected endpoints and boundaries of piecewise functions. The rigorous analysis of the exercise's piecewise function teaches the valuable lesson of examining every aspect of a function's behavior to conclusively determine the presence or absence of local extrema.