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Chapter 12: Problem 46

Your other friend tells you that she has found a continuous function with two critical points, one a relative minimum and one a relative maximum, and no point of inflection between them. Can she be right?

Short Answer

Expert verified
No, your friend cannot be right. In a continuous function with two critical points (one relative minimum and one relative maximum), there must be a point of inflection between them, as the concavity of the function needs to change for the derivative to change its sign.

Step by step solution

01

Recall definitions

A critical point of a function is a point where its derivative is either zero or does not exist. A relative minimum is a point where a function has a locally smallest value. A relative maximum is a point where a function has a locally largest value. A point of inflection is a point where the concavity of a function changes, meaning the second derivative is either zero or does not exist and changes its sign around the point.
02

Consider the First and Second Derivative

Suppose the function is \(f(x)\), and the two critical points are \(x_1\) and \(x_2\). At a relative minimum, the derivative \(f'(x)\) changes from negative to positive. At a relative maximum, the derivative \(f'(x)\) changes from positive to negative. Furthermore, at the point of inflection, the second derivative \(f''(x)\) changes its sign. Let's say x1 is the relative minimum and x2 is the relative maximum. Then between x1 and x2, the function needs to be concave up and concave down at the same time, considering that its first derivative changes sign. This would imply that the function has a point of inflection between these two critical points.
03

Conclusion

Based on the analysis above, a continuous function with two critical points (one being a relative minimum and the other one a relative maximum) must have a point of inflection between them. Therefore, your friend's claims are incorrect.

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Key Concepts

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

Continuous Function
Understanding what a continuous function means is crucial in calculus. In essence, a function is continuous if you can draw its graph without lifting your pencil. This implies that the function has no breaks, jumps, or holes. In mathematical terms, a function \( f(x) \) is continuous at a point \( c \) if:
  • \( f(c) \) is defined.
  • The limit as \( x \) approaches \( c \) of \( f(x) \) exists.
  • The limit as \( x \) approaches \( c \) of \( f(x) \) equals \( f(c) \).
Beyond individual points, a function is continuous over an interval if it is continuous at every point within that interval.
When dealing with critical points or analyzing the behavior of functions, continuity ensures that the function behaves predictably. Without continuity, critical points can result in undefined or unexpected outcomes.
Critical Points
Critical points are essential in determining where a function might change its increasing or decreasing behavior. These points occur where the derivative of the function is zero or does not exist.
For a given function \( f(x) \), its critical points, \( x_1, x_2, \ldots \), are solutions to the equation \( f'(x) = 0 \) or where \( f'(x) \) does not exist.
  • If a function has a derivative that equals zero, it means that tangent at that point is horizontal, which indicates a potential maximum or minimum.
  • Alternatively, if a function's derivative is undefined at a point, it might signify a cusp or vertical tangent.
It's important to note that not all critical points are points of maximum or minimum. Some could simply be flat points that don鈥檛 exhibit any local extremum behavior. Identifying whether a critical point is a relative minimum, maximum, or neither, involves further analysis such as the second derivative test.
Relative Minimum and Maximum
In calculus, identifying relative minimum and maximum points helps us understand the regions where a function achieves its lowest or highest local values.
A relative minimum is a point \( x_1 \) where the function \( f(x) \) is lower than any other nearby points, making it a 'dip' in the graph. Conversely, a relative maximum is at \( x_2 \), where \( f(x) \) is higher than the surrounding points, creating a 'peak'.
  • A relative minimum occurs when \( f'(x) \) changes from negative to positive. Here, the function transitions from decreasing to increasing.
  • A relative maximum occurs when \( f'(x) \) changes from positive to negative, indicating a shift from increasing to decreasing behavior.
Utilizing the second derivative \( f''(x) \), we can further determine the concavity of the function at these points. If \( f''(x) > 0 \), the function is concave up, confirming a local minimum, and if \( f''(x) < 0 \), it is concave down, confirming a local maximum.

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Most popular questions from this chapter

The following graph shows the approximate value of the United States Consumer Price Index (CPI) from December 2006 through July \(2007.31\) The approximating curve shown on the figure is given by $$I(t)=-0.04 t^{3}+0.4 t^{2}+0.1 t+202 \quad(0 \leq t \leq 7)$$ where \(t\) is time in months ( \(t=0\) represents December 2006). a. Use the model to estimate the monthly inflation rate in February \(2007(t=2) .\) [Recall that the inflation rate is \(\left.I^{\prime}(t) / I(t) .\right]\) b. Was inflation slowing or speeding up in February \(2007 ?\) c. When was inflation speeding up? When was inflation slowing? HINT [See Example 3.]

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