Problem 37 - Calculus

Question

(a) Sketch the graph of a continuous function \(f\) with all of the following properties: (i) \(f(0)=2\) (ii) \(f(x)\) is decreasing for \(0 \leq x \leq 3\) (iii) \(f(x)\) is increasing for \(35\) (v) \(f(x) \rightarrow 9\) as \(x \rightarrow \infty\) (b) Is it possible that the graph of \(f\) is concave down for all \(x>6 ?\) Explain.

Step-by-Step Solution

Answer

Yes, the graph can be concave down for all \(x > 6\).

Step 1: Identify Key Points

Let's mark the key points on the graph based on the given properties. We know that the function must pass through the point \((0, 2)\) because given is \(f(0)=2\). The graph is decreasing on \([0, 3]\), begins increasing just after \(x=3\) until \(x=5\), and decreases again for \(x>5\).

Step 2: Determine End Behavior

According to the property: \(f(x) \rightarrow 9\) as \(x \rightarrow \infty\), the function should approach a horizontal asymptote at \(y = 9\) as \(x\) increases.

Step 3: Sketch the Graph

On a coordinate plane, plot the point \((0, 2)\). From there, draw a decreasing curve until \(x=3\), where it should reach a local minimum. Then, draw an increasing curve that peaks at \(x=5\). After \(x=5\), the graph should decrease gradually, leveling off to approach the line \(y=9\) as \(x\) becomes very large.

Step 4: Assess Concavity for \(x > 6\)

For the concavity question, observe that even though the function is approaching its end behavior near a horizontal asymptote, it can remain concave down. A function can be concave down if its derivative is decreasing, even if the y-values themselves are increasing or leveling off.

Step 5: Conclusion for Part (b)

It is indeed possible for the graph of \(f\) to be concave down for all \(x > 6\). For example, a gently sloping curve that stops becoming less steep as it approaches \(y=9\) can be continuously concave down.

Key Concepts

Concavity

In mathematics, concavity refers to the curvature of a graph. For a function to be concave down in a region, its graph must resemble an "n" shape. This means as you move along the graph, it bends downwards.

Here’s how you can determine concavity:

• A function is concave down if its second derivative is negative in that interval.
• This implies the rate at which the slope (or first derivative) is changing is decreasing.

For the function given, being concave down for all \(x > 6\) means that it bends downward even as the function approaches the horizontal asymptote at \(y = 9\).

Even if a function’s values or slopes approach a limit, its concavity can still narrate another story of its shape.

Asymptote

The concept of an asymptote is crucial in understanding the behavior of graphs at their extreme ends. For a function, an asymptote is a line that the graph of the function approaches, but never quite touches.

In this exercise, the horizontal asymptote at \(y = 9\) indicates that as \(x\) becomes very large, \(f(x)\) gets closer and closer to 9.

• This tells us the function’s end behavior, which is essential for accurately sketching the graph.
• A horizontal asymptote does not prevent the graph from crossing it in other regions, but states the graph will tend towards the line as \(x\) goes to infinity.

Remember, horizontal asymptotes often represent limits at infinity, which is essential when considering the rate of change and overall shape of the function.

Graph Sketching

Sketching the graph of a continuous function provides a visual representation of its behavior. It incorporates properties like critical points, intervals of increase/decrease, concavity, and asymptotes.

When given conditions like those in the exercise, start by plotting known points. For instance:\(f(0) = 2\) gives you a specific starting point on the graph at \( (0, 2) \).

• Begin by drawing the curve based on where the function is increasing or decreasing, respecting the critical points (like \(x = 3\) and \(x = 5\)).
• Adjust your graph to show the correct approach toward the asymptote, ensuring the curve flattens towards \(y = 9\).
• Consider concavity to add detail between segments.

Always ensure each piece of information given is incorporated in the sketch to produce an accurate and informative representation of the function.