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Chapter 4: Problem 77

Critical pts. \(x=0\) and \(x=2\); local max at \(x=0\), local min at \(x=2\)

Short Answer

Expert verified
Question: Explain the concept of critical points and how they are used to determine whether a point is a local maximum, local minimum, or a saddle point given the critical points \(x = 0\) and \(x = 2\), where there is a local maximum at \(x = 0\) and a local minimum at \(x = 2\). Answer: Critical points are points where a function has either a local maximum, local minimum, or a saddle point. They occur when the derivative of the function is either zero or does not exist. To determine the nature of a critical point, we use the First Derivative Test. According to this test, if the derivative changes from positive to negative, the point is a local maximum, and if it changes from negative to positive, the point is a local minimum. If there is no sign change, the point is neither a maximum nor a minimum (a saddle point). In the given problem, the critical points \(x = 0\) and \(x = 2\) are confirmed as a local maximum and local minimum, respectively, due to the change in the signs of the derivative around these points.

Step by step solution

01

Introduction to Critical Points and First Derivative Test

Critical points are points where the function has either a local maximum, local minimum, or a saddle point. A point on a curve is considered a critical point when the derivative (the slope) of the function is either zero or does not exist. To find out if a critical point is a local maximum or local minimum, we can use the First Derivative Test. In this test, if the derivative changes from positive to negative at a critical point, that point corresponds to a local maximum. If it changes from negative to positive, the point is a local minimum. If there is no sign change, it's neither a maximum nor a minimum (a saddle point).
02

Differentiating the function

Determine the derivative of the given function, which will help us examine the critical points. Since we are not provided with the function in this exercise, we will illustrate the concept using a generic function. Let \(f(x)\) be the function. Find the first derivative \(f'(x)\).
03

Identify the critical points

We are given the critical points: \(x = 0\) and \(x = 2\). Additionally, we know that at \(x = 0\) we have a local maximum, and at \(x = 2\) we have a local minimum. To verify this, we can examine the signs of the derivative around these points according to the First Derivative Test mentioned in Step 1.
04

Verify the local maximum at \(x = 0\)

To check if there is a local maximum at \(x = 0\), we will examine the signs of \(f'(x)\) around this point. Pick any point that is less than \(0\) and find the value of \(f'(x)\) at that point. If it's positive, we can confirm that the slope is positive before \(x = 0\). Similarly, pick a point greater than \(0\) and find the value of \(f'(x)\) at that point. If it's negative, we can confirm that the slope is negative after \(x = 0\). This confirms a change in the sign of the derivative from positive to negative, and hence there is a local maximum at \(x = 0\).
05

Verify the local minimum at \(x = 2\)

To check if there is a local minimum at \(x = 2\), we will examine the signs of \(f'(x)\) around this point. Pick any point that is less than \(2\) and find the value of \(f'(x)\) at that point. If it's negative, we can confirm that the slope is negative before \(x = 2\). Similarly, pick a point greater than \(2\) and find the value of \(f'(x)\) at that point. If it's positive, we can confirm that the slope is positive after \(x = 2\). This confirms a change in the sign of the derivative from negative to positive, and hence there is a local minimum at \(x = 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Derivative Test
The First Derivative Test is a powerful tool in calculus used to identify the nature of critical points of a function. Critical points occur where the derivative of the function, denoted as \( f'(x) \), is either zero or undefined.
To employ the First Derivative Test, follow these steps:
  • Calculate the first derivative, \( f'(x) \), of the function.
  • Identify critical points by setting \( f'(x) = 0 \) or finding where \( f'(x) \) does not exist.
  • Analyze the sign changes of the derivative around each critical point:
    • If the derivative changes from positive to negative as it passes through a critical point, that critical point is a local maximum.
    • If the derivative changes from negative to positive, the critical point is a local minimum.
    • If there is no change in the sign of \( f'(x) \), the critical point is not an extreme value. It could be a saddle point or a point of inflection.
This test provides a straightforward method to assess where a function increases or decreases and helps in sketching the behavior of graphs.
Local Maximum
A local maximum on a function is a point where the function's value is higher than all nearby values. In simpler terms, it is a small peak on the graph.
For a critical point \( x_0 \) to be a local maximum, the derivative \( f'(x) \) should change from positive to negative as \( x \) moves through \( x_0 \).
To verify a local maximum, follow these steps:
  • Locate the critical point where \( f'(x) = 0 \) or \( f'(x) \) does not exist.
  • Choose a value less than \( x_0 \) and evaluate \( f'(x) \). If the result is positive, the function is increasing before \( x_0 \).
  • Choose a value greater than \( x_0 \) and evaluate \( f'(x) \). If the result is negative, the function is decreasing after \( x_0 \).
At the local maximum \( x_0 \), the function transitions from increasing to decreasing, confirming that \( x_0 \) is indeed a peak point.
Local Minimum
A local minimum is essentially the opposite of a local maximum. It is a point where the function's value is lower than all nearby values, like a small trough on the graph.
In a local minimum, the derivative \( f'(x) \) changes from negative to positive as \( x \) crosses the critical point \( x_0 \).
The process to verify a local minimum involves:
  • Identifying the critical point where \( f'(x) = 0 \) or \( f'(x) \) is undefined.
  • Selecting a value less than \( x_0 \) and showing that \( f'(x) \) is negative, confirming the function decreases before \( x_0 \).
  • Choosing a value greater than \( x_0 \) and proving that \( f'(x) \) is positive, signifying that the function increases after \( x_0 \).
Thus, at a local minimum, the function transitions from decreasing to increasing, creating a low point or valley on the function graph.
Saddle Point
A saddle point is a critical point on a graph that is not a local maximum nor a local minimum. The graph appears to "sit" at this point like a saddle, sometimes sloping upwards in one direction and downwards in another.
At a saddle point, the First Derivative Test shows no sign change of \( f'(x) \) as \( x \) passes through the critical point.
To identify a saddle point:
  • Find when \( f'(x) = 0 \) or \( f'(x) \) is not defined; these are your critical points.
  • Check the derivative on either side of these critical points.
  • If the sign of the derivative does not change, the critical point is a saddle point.
Saddle points are important in calculus for analyzing multi-dimensional graphs and determine where the function might not have peaks or troughs but still exhibit interesting behavior.

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