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Chapter 4: Problem 65

Explain what is wrong with the statement. If \(p\) is a critical point, and \(f^{\prime}\) is negative to the left of \(p\) and positive to the right of \(p,\) and if \(f^{\prime \prime}(p)\) exists, then \(f^{\prime \prime}(p)>0\).

Short Answer

Expert verified
The statement incorrectly assumes \( f''(p) > 0 \) from \( f' \) sign change without evaluating \( f''(p) \).

Step by step solution

01

Understanding Critical Points

A critical point of a function occurs where the derivative is zero or undefined. It is a potential location for a local maximum, minimum, or an inflection point. For this problem, we need to analyze what having a positive derivative to the right and a negative to the left of a critical point implies.
02

Analyzing Derivative Changes

If the derivative of a function, \( f' \), is negative to the left of a point \( p \) and positive to the right, it indicates that \( p \) is likely a local minimum. This information doesn't directly relate to the sign of the second derivative at \( p \).
03

Second Derivative Test

The second derivative test states that if \( f''(p) > 0 \), then there is a local minimum at \( p \), and if \( f''(p) < 0 \), there is a local maximum. If \( f''(p) = 0 \), the test is inconclusive.
04

Identifying the Error

The error in the statement is the implication that \( f''(p) > 0 \) solely from the behavior of \( f'(x) \) around \( p \). The change in sign of \( f'(x) \) indicates a local minimum but does not imply that \( f''(p) > 0 \) without directly evaluating \( f''(p) \). For example, \( f'(x) = (x-p)^3 \) has \( f'(x) < 0 \) for \( x < p \) and \( f'(x) > 0 \) for \( x > p \) but still satisfies \( f''(p) = 0 \).

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Key Concepts

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

Derivative
The derivative of a function, denoted as \( f'(x) \), represents the rate at which the function's value changes as \( x \) changes. It's essentially the slope of the tangent line at any given point on a curve. This makes it a powerful tool for understanding the behavior of functions.
  • When \( f'(x) > 0 \), the function is increasing at that point.
  • When \( f'(x) < 0 \), the function is decreasing.
  • \( f'(x) = 0 \) often indicates a critical point, where the function could have a local maximum, minimum, or a saddle point.
In our specific problem, the derivative changes from negative to positive at the critical point \( p \), indicating a likely change in the direction of the curve at this point. This information only tells us about the movement rather than the shape of the curve.
Second Derivative Test
The second derivative, \( f''(x) \), gives us information about the concavity of the function at a particular point. It directly tells us about the "curviness" of the function.
  • If \( f''(x) > 0 \), the function is concave up, like a smile, indicating a local minimum might exist.
  • If \( f''(x) < 0 \), the function is concave down, like a frown, indicating a local maximum might exist.
  • When \( f''(x) = 0 \), the test does not provide definitive information about maxima or minima.
In the original statement evaluated, the mistake comes from assuming that a change in sign of the first derivative automatically means \( f''(p) > 0 \). However, \( f''(p) \) must be directly calculated to confirm concavity and determine the presence of a local minimum or maximum.
Local Minimum
A local minimum of a function occurs at a point where the function changes direction from decreasing to increasing. Visually, it is a point where the curve reaches its lowest value in a small neighborhood around it.
To mathematically identify a local minimum at a critical point:\
  • The first derivative \( f'(x) \) changes from negative to positive around a critical point.
  • The second derivative \( f''(p) \) is positive, confirming the presence of a concave up curve.
In the problem presented, the first derivative changes sign at \( p \), suggesting \( p \) as a local minimum. However, without verifying \( f''(p) > 0 \), this conclusion is incomplete. Hence, always calculating the second derivative ensures accuracy.
Inflection Point
An inflection point is where a function changes concavity, from concave up to concave down or vice versa. It's different from the critical points that indicate local maxima or minima.
Characteristics of an inflection point include:
  • The second derivative \( f''(x) \) equals zero (\( f''(x) = 0 \)) at the inflection point itself.
  • A sign change must occur in \( f''(x) \) before and after the point.
In the exercise, the presence of an inflection point might confuse with local extrema, as the first derivative can change signs while the second derivative might be zero. Therefore, carefully analyzing \( f''(x) \) around potential inflection points is crucial to distinguish them from true local extrema.

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Most popular questions from this chapter

In a romantic relationship between Angela and Brian, who are unsuited for each other, \(a(t)\) represents the affection Angela has for Brian at time \(t\) days after they meet, while \(b(t)\) represents the affection Brian has for Angela at time \(t .\) If \(a(t)>0\) then Angela likes Brian; if \(a(t)<0\) then Angela dislikes Brian; if \(a(t)=0\) then \(\mathrm{An}-\) gela neither likes nor dislikes Brian. Their affection for each other is given by the relation \((a(t))^{2}+(b(t))^{2}=c\) where \(c\) is a constant. (a) Show that \(a(t) \cdot a^{\prime}(t)=-b(t) \cdot b^{\prime}(t)\) (b) At any time during their relationship, the rate per day at which Brian's affection for Angela changes is \(b^{\prime}(t)=-a(t) .\) Explain what this means if Angela (i) Likes Brian, (ii) Dislikes Brian. (c) Use parts (a) and (b) to show that \(a^{\prime}(t)=b(t) .\) Explain what this means if Brian (i) Likes Angela. (ii) Dislikes Angela. (d) If \(a(0)=1\) and \(b(0)=1\) who first dislikes the other?

(a) Explain how you know that the following two pairs of equations parameterize the same line: $$x=2+t, y=4+3 t \text { and } x=1-2 t, y=1-6 t.$$ (b) What are the slope and \(y\) intercept of this line?

Describe the form of the limit \((0 / 0\) \(\left.\infty / \infty, \infty \cdot 0, \infty-\infty, 1^{\infty}, 0^{0}, \infty^{0}, \text { or none of these }\right) .\) Does I'Hopital's rule apply? If so, explain how. $$\lim _{x \rightarrow 0}(1+x)^{x}$$

Evaluate the limit using the fact that $$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e$$ $$\lim _{n \rightarrow \infty}\left(1+\frac{2}{n}\right)^{n}$$

Determine whether the limit exists, and where possible evaluate it. $$\lim _{n \rightarrow \infty} \sqrt[n]{n}$$
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