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Chapter 4: Problem 42

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

Short Answer

Expert verified
The graph of the function \(f(x)\) which fulfills the conditions is the piecewise function where \(f(x) = -x^2\) for \(x < 0\) and \(f(x) = x^2\) for \(x > 0\).

Step by step solution

01

Understand local minimum and maximum

A local minimum is a point in the function where the value of the variable is lower than or equal to the values at nearby points. Conversely, a local maximum is a point where the variable is larger than or equal to values at nearby points.
02

Sketch function with a local maximum

First, sketch a function with a local maximum. This could be simple, for example \(f(x) = -x^2\). The peak of this parabolic function happens at \(x=0\) and is a local maximum.
03

Sketch function with a higher local minimum

Now, we must sketch a function with a local minimum that is higher than the previous maximum. Consider a piecewise function merged with the previously drawn function. A suitable function could be \(f(x) = x^2\) for \(x > 0\). This function has a local minimum at \(x=0\) and given that we define it for \(x > 0\), it is always larger than the local maximum of the other function.
04

Merge and sketch the final graph

Now, merge these two functions into a piecewise function. Define \(f(x) = -x^2\) for \(x < 0\) and \(f(x) = x^2\) for \(x > 0\). This function is still differentiable and fulfills our conditions.

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Key Concepts

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

Local Minimum
A local minimum in calculus and mathematics is an important aspect of optimizing functions. It refers to a point on the graph of a function where the function value is lower than the value at any nearby point. Imagine climbing down a gentle slope and reaching a small dip or valley. This dip is the local minimum.
To visualize a local minimum, consider the function \(f(x) = x^2\) where at \(x=0\) the graph of this function is at its lowest point compared to points nearby. The function value is zero at this point, and values around it are higher. It鈥檚 important to note that a local minimum is not necessarily the lowest point on the entire graph鈥攖hat would be a global minimum.
  • Occurs at a point in the function
  • Lower than values right next to it
  • Not always the smallest overall value
Local Maximum
A local maximum is the opposite of a local minimum. It occurs at a point where the function value is higher than the values around it. Picture climbing the same slope again, but this time reaching the top of a hill. This hilltop is your local maximum.
For example, if you look at the function \(f(x) = -x^2\), the peak occurs at \(x = 0\). Here, the value of the function is higher compared to values nearby. It鈥檚 useful to understand these peaks when determining where functions increase and decrease locally.
  • Point where function value is higher than neighboring points
  • Like a small peak or hill on a mountain range
  • Not necessarily the highest value globally
Piecewise Function
A piecewise function is a versatile tool in mathematics that allows us to represent a function using different expressions based on the input value or range of input values. It combines multiple sub-functions into one, allowing for more complex forms and behaviors.
Think of a piecewise function like a chameleon, changing its form based on which segment of the domain you鈥檙e looking at. In the example from the exercise, the piecewise function is defined as \(f(x) = -x^2\) for \(x < 0\), and \(f(x) = x^2\) for \(x > 0\). This lets the function behave differently in different regions鈥攕howing a local maximum on one side and a local minimum on the other.
  • Involves multiple sub-functions combined together
  • Each sub-function applies to a specific interval of the domain
  • Useful for modeling situations with sudden changes or breaks

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Most popular questions from this chapter

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

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.]

Production Level Suppose \(c(x)=x^{3}-20 x^{2}+20,000 x\) is the cost of manufacturing \(x\) items. Find a production level that will minimize the average cost of making \(x\) items.

Tolerance The height and radius of a right circular cylinder are equal, so the cylinder's volume is \(V=\pi h^{3} .\) The volume is to be calculated with an error of no more than 1\(\%\) of the true value. Find approximately the greatest error that can be tolerated in the measurement of \(h,\) expressed as a percentage of \(h .\)

Group Activity Cardiac Output In the late 1860 s, Adolf Fick, a professor of physiology in the Faculty of Medicine in Wurtzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a minute. Your cardiac output as you read this sentence is probably about 7 liters a minute. At rest it is likely to be a bit under 6 \(\mathrm{L} / \mathrm{min}\) . If you are a trained marathon runner running a marathon, your cardiac output can be as high as 30 \(\mathrm{L} / \mathrm{min.}\) Your cardiac output can be calculated with the formula $$$=\frac{Q}{D}$$ where \(Q\) is the number of milliliters of \(\mathrm{CO}_{2}\) you exhale in a minute and \(D\) is the difference between the \(\mathrm{CO}_{2}\) concentration \((\mathrm{mL} / \mathrm{L})\) in the blood pumped to the lungs and the \(\mathrm{CO}_{2}\) concentration in the blood returning from the lungs. With \(Q=233 \mathrm{mL} / \mathrm{min}\) and \(D=97-56=41 \mathrm{mL} / \mathrm{L}\) $$y=\frac{233 \mathrm{mL} / \mathrm{min}}{41 \mathrm{mL} / \mathrm{L}} \approx 5.68 \mathrm{L} / \mathrm{min}$$ fairly close to the 6 \(\mathrm{L} / \mathrm{min}\) that most people have at basal (resting) conditions. (Data courtesy of J. Kenneth Herd, M.D. Quillan College of Medicine, East Tennessee State University.) Suppose that when \(Q=233\) and \(D=41,\) we also know that \(D\) is decreasing at the rate of 2 units a minute but that \(Q\) remains unchanged. What is happening to the cardiac output?
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