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Chapter 4: Problem 55

Give an example of: A function which has no critical points on the interval between 0 and 1

Short Answer

Expert verified
A function \( f(x) = e^x \) has no critical points between 0 and 1.

Step by step solution

01

Understand Critical Points

A critical point of a function occurs where its derivative is either zero or undefined. Therefore, we need to examine a function that does not meet these criteria on the interval between 0 and 1.
02

Choose a Suitable Function

Select a function known for having no critical points in a given interval. The function \( f(x) = e^x \) is a good candidate because its derivative, \( f'(x) = e^x \), is never zero or undefined.
03

Differentiate the Function

Compute the derivative of the function \( f(x) = e^x \). The derivative is \( f'(x) = e^x \), which is positive for all real numbers.
04

Analyze the Derivative

On the interval from 0 to 1, the derivative \( f'(x) = e^x \) is positive and never zero or undefined. Thus, the function \( f(x) = e^x \) has no critical points on this interval.

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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 gives us critical insight into how the function behaves. It tells us the rate at which the function's value changes as its input changes. Critical points occur where this rate of change, the derivative, is zero or undefined.
For instance, if we have a function like \( f(x) = e^x \), it鈥檚 crucial first to understand how its derivative behaves. Differentiating \( e^x \) gives us \( f'(x) = e^x \) again.
  • Important here is noticing that \( e^x \) is never zero or undefined because the exponential function is always positive.
  • This means there is no point in the interval where the function "stops" increasing or decreasing.
  • Therefore, there are no critical points for \( f(x) = e^x \) on any interval, including from 0 to 1.
Interval Analysis
Interval analysis involves examining how a function behaves over a specific range of values. This is particularly useful for understanding where critical points might or might not exist within an interval.
To analyze the interval from 0 to 1 for a function like \( f(x) = e^x \), we focus on its derivative, \( f'(x) = e^x \).
  • Since \( e^x \) is positive for all real numbers, the function is always increasing within the interval.
  • Because \( f'(x) \) never equals zero or becomes undefined, \( f(x) = e^x \) lacks any critical points.
  • This means that within the interval from 0 to 1, the function continuously increases at an exponential rate without any halts or reversals.
Exponential Function
Exponential functions like \( e^x \) have unique properties that make analyzing them straightforward, especially on certain intervals.
An exponential function is defined by a constant base raised to a variable exponent. In the case of \( e^x \), the base is Euler鈥檚 number \( e \), an irrational constant approximately equal to 2.71828.
  • The exponential function \( e^x \) is always positive and never zero. This makes its derivative \( e^x \) follow the same rule.
  • On any interval, including between 0 and 1, the function is consistently increasing, making it quite predictable in its behavior.
  • Due to its continuous growth, an exponential function like \( e^x \) lacks stops or pauses, hence having no critical points on any interval.

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Most popular questions from this chapter

What are the dimensions of the closed cylindrical can that has surface area 280 square centimeters and contains the maximum volume?

Investigate the one-parameter family of functions. Assume that \(a\) is positive. (a) Graph \(f(x)\) using three different values for \(a\) (b) Using your graph in part (a), describe the critical points of \(f\) and how they appear to move as \(a\) increases. (c) Find a formula for the \(x\) -coordinates of the critical point(s) of \(f\) in terms of \(a\) $$f(x)=(x-a)^{2}$$

Table 4.2 shows cost, \(C(q),\) and revenue, \(R(q)\) (a) At approximately what production level, \(q\), is profit maximized? Explain your reasoning. (b) What is the price of the product? (c) What are the fixed costs? $$\begin{array}{l|r|r|r|r|r|r|r}\hline q & 0 & 500 & 1000 & 1500 & 2000 & 2500 & 3000 \\\\\hline R(q) & 0 & 1500 & 3000 & 4500 & 6000 & 7500 & 9000 \\\\\hline C(q) & 3000 & 3800 & 4200 & 4500 & 4800 & 5500 & 7400 \\\\\hline\end{array}$$

Give an example of a function \(f\) that makes the statement true, or say why such an example is impossible. Assume that \(f^{\prime \prime}\) exists everywhere. \(f\) is concave up and \(f(x)\) is negative for all \(x\).

Table 4.3 shows marginal cost, \(M C,\) and marginal revenue, \(M R\) (a) Use the marginal cost and marginal revenue at a production of \(q=5000\) to determine whether production should be increased or decreased from \(5000 .\) (b) Estimate the production level that maximizes profit. $$\begin{array}{c|r|r|r|r|r|r}\hline q & 5000 & 6000 & 7000 & 8000 & 9000 & 10000 \\\\\hline M R & 60 & 58 & 56 & 55 & 54 & 53 \\\\\hline M C & 48 & 52 & 54 & 55 & 58 & 63 \\ \hline\end{array}$$
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