Problem 39 - Calculus

Question

Sketch the graph of one continuous function \(f\) that has all of the indicated properties. $$\begin{aligned}&f^{\prime}(x)>0 \text { for } 0< x<3 \text { and } 8< x<12\\\&f^{\prime}(x)<0 \text { for } 3< x<8 \end{aligned}$$ $$\begin{aligned}&f^{\prime \prime}(x)>0 \text { for } 5< x<9\\\&f^{\prime \prime}(x)<0 \text { for } 0< x<5 \text { and } 9< x<12\end{aligned}$$ $$\lim _{x \rightarrow \infty} f(x)=2$$

Step-by-Step Solution

Answer

Draw a curve increasing on (0, 3) and (8, 12), decreasing on (3, 8), concave down on (0, 5) and (9, 12), concave up on (5, 9). Approach y=2 as x increases.

Step 1: Determine Critical Points

First, identify the intervals where the derivative changes sign. Since \(f'(x) > 0\) for \(0 < x < 3\) and \(8 < x < 12\), and \(f'(x) < 0\) for \(3 < x < 8\), it indicates that there are critical points at \(x = 3\) and \(x = 8\). At these points, the graph changes from increasing to decreasing or vice versa.

Step 2: Analyze Concavity

Next, take the second derivative conditions into account: \(f''(x) < 0\) for \(0 < x < 5\) and \(9 < x < 12\), and \(f''(x) > 0\) for \(5 < x < 9\). This suggests concave down behavior in the intervals \((0, 5)\) and \((9, 12)\) and concave up behavior in the interval \((5, 9)\). At \(x = 5\) and \(x = 9\), there are points of inflection where the concavity changes.

Step 3: Sketch Increasing and Decreasing Intervals

Using the derivative information, sketch an increasing curve from \(x = 0\) to \(x = 3\) and from \(x = 8\) to \(x = 12\). Draw a decreasing section between \(x = 3\) and \(x = 8\).

Step 4: Sketch Concave Up and Down Segments

Incorporate the concavity into the sketch: draw the segment from \(x = 0\) to \(x = 5\) concave down, then switch to concave up from \(x = 5\) to \(x = 9\), and finally concave down from \(x = 9\) to \(x = 12\). Ensure that the changes in concavity appear at the appropriate points of inflection at \(x = 5\) and \(x = 9\).

Step 5: Check End Behavior

Lastly, note that \(\lim _{x \rightarrow \infty} f(x) = 2\). This means that as \(x\) approaches infinity, the function should approach the horizontal line \(y = 2\). Adjust the end behavior of your sketch accordingly to flatten out and approach \(y = 2\) from \(x = 12\) onward.

Key Concepts

Continuous Functions

Continuous functions are the fundamental building blocks in calculus and mathematical analysis that smoothly connect points along their curves without any interruption. In simple terms, a continuous function has no jumps, holes, or gaps.
For example, when we say that a function is continuous over an interval, it means you can draw its graph from one point to another without lifting your pen. Some important features of continuous functions include:

• Existence of Limits: For a function to be continuous at a point, the limit as you approach the point must equal the function’s value at that point.
• Smooth Graph: The graph of a continuous function does not have any breaks or sharp corners.

Understanding continuity is essential when sketching graphs since it ensures that the plotted function behaves as expected across its domain.

Critical Points

Critical points on the graph of a function occur where the first derivative is zero or undefined. These points are significant because they represent where a change in increasing or decreasing behavior happens, or where a function might reach a peak or a valley.
In the problem, the function has critical points at

• \(x = 3\) and
• \(x = 8\).

These points are determined by changes in the sign of the derivative,\( f'(x)\), which indicates that at these points the function changes from increasing to decreasing or vice versa. Checking these points helps identify potential local maxima or minima in the graph.
When sketching functions, understanding critical points allows you to capture the essential shape and major turns of the graph.

Concavity

Concavity provides insight into the curvature of the graph of a function. By examining the second derivative, \(f''(x)\), we can determine whether a function is concave up or concave down in certain intervals.
A function is:

• Concave Up where \(f''(x) > 0\): This looks like a cup, meaning the function is curving upwards and any tangent line will lie below the graph.
• Concave Down where \(f''(x) < 0\): This is like a cap, indicating the function is curving downwards with tangent lines above the graph.

For points

• \(x = 5\)
• \(x = 9\)

in the exercise, we find inflection points where concavity changes.
Understanding concavity and points of inflection aid in giving the graph its correct shape and helping us understand the overall structure of the function.

End Behavior

The end behavior of a function describes what happens to the values of the function as the input, \(x\), grows very large or very small. This is crucial for accurately extending the curve to infinity or negative infinity in sketches.
In the exercise, one of the given conditions is that \(\lim_{x \to \infty} f(x) = 2\). This indicates that as \(x\) approaches infinity, the function levels out or approaches the horizontal line \(y = 2\).
Understanding end behavior is vital when sketching graphs as it impacts the tail of the curve, ensuring it behaves properly at extreme values of \(x\).
This flattening out towards a horizontal asymptote is often a signature of polynomial or rational functions where other terms diminish in magnitude.