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Chapter 2: Problem 49

Sketch a graph that possesses the characteristics listed. Answers may vary. \(f\) is increasing and concave down on \((-\infty, 1)\) fis increasing and concave up on \((1, \infty)\)

Short Answer

Expert verified
Draw an increasing curve concave down before x = 1, and concave up after x = 1.

Step by step solution

01

- Understand the Characteristics

The function, denoted by f, has specific behaviors in different intervals. In \((-\infty, 1)\), f is increasing and concave down. In \((1, \infty)\), f is increasing and concave up.
02

- Identify Concave Down Region

When a function is concave down, it opens downward like an upside-down bowl. From \( -\infty \) to 1, f is both increasing and concave down. Draw a curve that rises gradually towards x = 1 while maintaining a downward curvature.
03

- Identify Concave Up Region

When a function is concave up, it opens upwards like a bowl. From 1 to \(\infty\), f is both increasing and concave up. Continue the curve past x = 1, allowing it to rise more steeply while curving upwards.
04

- Sketch the Graph

Combine the identified behaviors into a single continuous graph. Ensure that the curve transitions smoothly from being concave down before x = 1 to being concave up after x = 1. The result should depict a function that increases throughout, with a change in concavity at x = 1.

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Key Concepts

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

Increasing Function
An increasing function is a function where, as the x-value increases, the y-value also increases. In mathematical terms, for any two points, \(x_1 < x_2\), the corresponding y-values will satisfy \(f(x_1) < f(x_2)\). This means that the function never decreases; it always goes up as we move from left to right along the x-axis.

In the given exercise, we have to sketch a function f that is increasing. Specifically, it is increasing on two intervals: \((-\rightarrow, 1)\) and \( (1, \rightarrow)\). This means the function continuously rises, regardless of whether it is concave up or concave down. When sketching such a graph, always make sure there are no downward slopes.

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Most popular questions from this chapter

Total revenue, cost, and profit. Using the same set of axes. sketch the graphs of the total-revenue, total-cost, and total. profit functions. $$R(x)=50 x-0.5 x^{2}, \quad C(x)=10 x+3$$

U.S. oil production. One model of oil production in the United States is given by $$ \begin{aligned} P(t)=& 0.0000000219 t^{4}-0.0000167 t^{3}+0.00155 t^{2} \\ &+0.002 t+0.22, \quad 0 \leq t \leq 110 \end{aligned} $$ where \(P(t)\) is the number of barrels of oil, in billions, produced in a year, \(t\) years after \(1910 .\) (Source: Beyond Oil, by Kenneth S. Deffeyes, p. \(41,\) Hill and Wang, New York, \(2005 .)\) a) According to this model, what is the absolute maximum amount of oil produced in the United States and in what year did that production occur? b) According to this model, at what rate was United States oil production declining in 2004 and in \(2010 ?\)

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