Problem 40 - Calculus

Question

Sketch the graph of one continuous function \(f\) that has all of the indicated properties. $$\begin{aligned}&f^{\prime}(x)>0 \text { for }-2< x<2\\\&f^{\prime}(x)<0 \text { for }-6< x<-2 \text { and } 2< x<6\end{aligned}$$ $$\begin{aligned}&f^{\prime \prime}(x)>0 \text { for }-3< x<0 \text { and } 3< x<6\\\&f^{\prime \prime}(x)<0 \text { for }-6< x<-3 \text { and } 0< x<3\end{aligned}$$ $$\lim _{x \rightarrow-\infty} f(x)=2 \text { and } \lim _{x \rightarrow \infty} f(x)=3$$

Step-by-Step Solution

Answer

Sketch from -6 to 6 with increasing/decreasing and concavity as described; horizontal asymptotes at 2 and 3.

Step 1: Understand Increasing and Decreasing Intervals

The given conditions describe where the function is increasing or decreasing. If \( f'(x) > 0 \) for \(-2 < x < 2\), the function is increasing in this interval. If \( f'(x) < 0 \) for \(-6 < x < -2\) and \(2 < x < 6\), the function is decreasing in these intervals.

Step 2: Understand Concavity Intervals

The conditions on the second derivative tell us where the function is concave up or concave down. If \( f''(x) > 0 \) for \(-3 < x < 0\) and \(3 < x < 6\), the function is concave up in these intervals. If \( f''(x) < 0 \) for \(-6 < x < -3\) and \(0 < x < 3\), the function is concave down in these intervals.

Step 3: Behavior at Infinity

The limit conditions \( \lim_{x \to -\infty} f(x) = 2 \) and \( \lim_{x \to \infty} f(x) = 3 \) describe the horizontal asymptotes of the function as \(x\) approaches negative and positive infinity, respectively.

Step 4: Sketch the Function

Start plotting the graph using the intervals and behavior identified:- From \(-6 < x < -3\), the function is decreasing and concave down.- From \(-3 < x < -2\), the function is decreasing and concave up.- From \(-2 < x < 0\), the function is increasing and concave up.- From \(0 < x < 2\), the function is increasing and concave down.- From \(2 < x < 3\), the function is decreasing and concave down.- From \(3 < x < 6\), the function is decreasing and concave up.Add horizontal asymptotes at \(y=2\) as \(x \rightarrow -\infty\) and \(y=3\) as \(x \rightarrow +\infty\).

Key Concepts

Increasing and Decreasing Intervals

In calculus, understanding increasing and decreasing intervals helps to grasp how a function behaves over various ranges. A function is said to be increasing on an interval if its derivative, denoted as \( f'(x) \), is greater than zero over that entire interval. Conversely, a function is decreasing if \( f'(x) \) is less than zero.- For the interval \(-6 < x < -2\), the function \( f \) is decreasing because \( f'(x) < 0 \).- For \(-2 < x < 2\), \( f \) is increasing since \( f'(x) > 0 \).- Lastly, from \(2 < x < 6\), \( f \) also decreases as \( f'(x) < 0 \).Knowing where a function increases or decreases allows you to predict the shape and direction of the graph without plotting every single point. It also informs about potential local maxima or minima, which occur at transitions from increasing to decreasing intervals, and vice versa.

Concavity Intervals

Concavity is another important aspect of a function's graph. It refers to the direction the curve opens. If the second derivative \( f''(x) \) is positive over an interval, the function is concave up, resembling a "cup" shape. If \( f''(x) \) is negative, the function is concave down, resembling an "arch".- From \(-6 < x < -3\), \( f''(x) < 0 \): the function is concave down.- From \(-3 < x < 0\), \( f''(x) > 0 \): the function is concave up.- From \(0 < x < 3\), \( f''(x) < 0 \): the function remains concave down.- Finally, from \(3 < x < 6\), \( f''(x) > 0 \): the function is concave up.Understanding concavity helps to visualize where a function might have inflection points—where concavity changes. Inflection points, while not always obvious on a graph, are critical for a comprehensive understanding of the function's changing behavior.

Horizontal Asymptotes

Horizontal asymptotes describe a line that a function approaches as \( x \) tends to positive or negative infinity. It is crucial in determining the end behavior of a function.For our function, the behavior is determined:- As \( x \rightarrow -\infty \), the function approaches the line \( y = 2 \). This suggests that no matter how far left you travel on the graph, the values of \( f(x) \) will closely approach 2.- As \( x \rightarrow \infty \), \( f(x) \) approaches the line \( y = 3 \). Hence, moving infinitely to the right, the function's values settle near 3.Establishing horizontal asymptotes is significant, especially when examining functions that stretch across a wide range of x-values. This helps provide boundaries within which the function's output lies, preventing misinterpretation of the graph's continuity and behavior far from the origin.