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Chapter 4: Problem 58

Multiple Choice If \(f(0)=f^{\prime}(0)=f^{n}(0)=0,\) which of the following must be true? \(\mathrm (A) There is a local maximum of \)f\( at the origin. (B) There is a local minimum of \)f\( at the origin. (C) There is no local extremum of \)f\( at the origin. (D) There is a point of inflection of the graph of \)f\( at the origin. (E) There is a horizontal tangent to the graph of \)f$ at the origin.

Short Answer

Expert verified
The correct option is (E). There is a horizontal tangent to the graph of \(f\) at the origin.

Step by step solution

01

Statement Analysis

The given information tells us that \(f(0)=f'(0)=f''(0)=0\). We need to verify, from the options given, which of them must be true based on what we know about the function and its derivatives. The options propose various possibilities of a local maximum, minimum, an inflection point or a horizontal tangent.
02

Understanding Derivatives

First, let's recall what we know about derivatives. The first derivative of a function at a point gives us the slope of the tangent line to the function at that point. A derivative equal to zero means that the function has a horizontal tangent at that point, indicating a potential maximum, minimum, or inflection point. The second derivative tells us about the concavity of the function. A function has an inflection point (changes concavity) where its second derivative is zero. However, in order for it to be an inflection point, the second derivative must change signs. If the second derivative does not change signs, the function does not actually have an inflection point, even though the second derivative is zero.
03

Using The Derivative Information

Given that \(f'(0)=0\), we know that the function has a horizontal tangent at \(x=0\), but we cannot determine whether it has a local maximum, local minimum, or point of inflection. Now, with \(f''(0)=0\), it suggests a possible point of inflection at \(x=0\), but we do not have enough information to determine if the concavity changes. Since we don't have enough information to clarify whether it's a local maximum, local minimum or an inflection point we can rule out options (A), (B), and (D).
04

Final Conclusion

After analyzing each option and applying our understanding of derivatives, we find that option (E) 'There is a horizontal tangent to the graph of \(f\) at the origin' must be true. The other options can't be determined only from the given information.

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Key Concepts

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

Derivatives
In calculus, derivatives are a fundamental concept that describes how a function changes as its input changes. The derivative of a function at a particular point gives the slope of the tangent line to the function at that point. This can inform us about how steeply the graph is increasing or decreasing at any given point.
  • The first derivative, often denoted as \( f'(x) \), tells us whether the function is increasing or decreasing.
  • If \( f'(x) = 0 \), this indicates a potential horizontal tangent, meaning the graph could have a stationary point there.
  • Higher-order derivatives, such as the second derivative \( f''(x) \), provide information about the curvature of the graph.
Derivatives also help identify critical points on the function where potential extrema (maximum or minimum values) could occur, but further analysis involving higher derivatives or the behavior of the derivative function can be necessary to draw conclusions about the nature of those points.
Local Extrema
Local extrema refer to points on a function where it takes on a local maximum or minimum value. These points are where the function changes direction from increasing to decreasing, or vice versa, in a neighborhood around the point.
  • A local maximum is a point \( x=a \) where the function value \( f(a) \) is greater than any function values in some small neighborhood around \( a \).
  • A local minimum is a point \( x=b \) where \( f(b) \) is less than any function values in a small neighborhood around \( b \).
To determine local extrema using derivatives:
  • Find where the first derivative \( f'(x) = 0 \) or is undefined, known as critical points.
  • Use the second derivative test, where \( f''(x) > 0 \) indicates a local minimum, and \( f''(x) < 0 \) indicates a local maximum.
If \( f''(x) = 0 \), this test is inconclusive, and further analysis is needed.
Inflection Points
An inflection point of a function is where the graph changes its concavity - that is, shifts from being concave up to concave down, or vice versa.
  • At an inflection point, the second derivative of the function \( f''(x) \) is zero. However, just because \( f''(x) = 0 \) does not guarantee an inflection point.
  • For a true inflection point, \( f''(x) \) must change sign. This means if the function was previously concave up (\( f''(x) > 0 \)) it should switch to concave down (\( f''(x) < 0 \)), or the opposite.
Identifying inflection points is crucial in understanding the behavior of a graph, as it tells us where the curvature changes and leads to more complex understanding beyond just extrema.
Horizontal Tangents
Horizontal tangents occur at points on a function where the slope of the tangent line is zero. This means the first derivative at that point is zero \( f'(x) = 0 \), suggesting that the rate of change of the function at that point is neither increasing nor decreasing.
  • A horizontal tangent indicates a potential maximum or minimum, as the function temporarily "flattens" out at the tangent point.
  • However, a horizontal tangent could also be found at an inflection point, where the function changes its concavity without being an extremum.
Finding horizontal tangents is often the first step in locating critical points in a function which need to be analyzed further to determine their nature. In many cases, then evaluating the second derivative can help clarify whether it's a point of local extremum or an inflection point.

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Most popular questions from this chapter

Group Activity In Exercises \(39-42,\) sketch a graph of a differentiable function \(y=f(x)\) that has the given properties. $$\begin{array}{l}{\text { A local minimum value that is greater than one of its local maxi- }} \\ {\text { mum values. }}\end{array}$$

True or False A continuous function on a closed interval must attain a maximum value on that interval. Justify your answer.

The Linearization is the Best Linear Approximation Suppose that \(y=f(x)\) is differentiable at \(x=a\) and that \(g(x)=m(x-a)+c(m\) and \(c\) constants). If the error \(E(x)=f(x)-g(x)\) were small enough near \(x=a,\) we might think of using \(g\) as a linear approximation of \(f\) instead of the linearization \(L(x)=f(a)+f^{\prime}(a)(x-a) .\) Show that if we impose on \(g\) the conditions i. \(E(a)=0\) ii. \(\lim _{x \rightarrow a} \frac{E(x)}{x-a}=0\) then \(g(x)=f(a)+f^{\prime}(a)(x-a) .\) Thus, the linearization gives the only linear approximation whose error is both zero at \(x=a\) and negligible in comparison with \((x-a)\) .

Formulas for Differentials Verify the following formulas. (a) \(d(c)=0(c\) a constant \()\) (b) \(d(c u)=c d u(c\) a constant \()\) (c) \(d(u+v)=d u+d v\) (d) \(d(u \cdot v)=u d v+v d u\) (e) \(d\left(\frac{u}{v}\right)=\frac{v d u-u d v}{v^{2}}\) (f) \(d\left(u^{n}\right)=n u^{n-1} d u\)

Multiples of \(P i\) Store any number as \(X\) in your calculator. Then enter the command \(X-\tan (X) \rightarrow X\) and press the ENTER key repeatedly until the displayed value stops changing. The result is always an integral multiple of \(\pi .\) Why is this so? [Hint: These are zeros of the sine function.]
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