Problem 54 - Essential Calculus

Question

Sketch a graph of a differentiable function \(f\) that satisfies the following conditions and has \(x=2\) as its only critical number. \(f^{\prime}(x)<0\) for \(x<2 \quad f^{\prime}(x)>0\) for \(x>2\) \(\lim _{x \rightarrow-\infty} f(x)=\lim _{x \rightarrow \infty} f(x)=6\)

Step-by-Step Solution

Answer

The graph of function \(f\) has a minimum at \(x=2\) and a horizontal asymptote at \(y=6\). To the left of \(x=2\), the graph is decreasing and to the right, the graph is increasing. Both sides trend towards \(y=6\) as \(x\) approaches infinity and negative infinity.

Step 1: Identify the Impact of given Derivative Conditions

The derivative's sign indicates the function's direction. If \(f^{\prime}(x)<0\), the function is decreasing and if \(f^{\prime}(x)>0\), the function is increasing. Hence, since \(f^{\prime}(x)<0\) for \(x<2\) and \(f^{\prime}(x)>0\) for \(x>2\), this implies that \(f\) is decreasing until \(x = 2\) and then starts increasing. Thus, the function has a minimum at \(x =2\).

Step 2: Use the Limit to Identify the Asymptote

Given that \(\lim _{x\rightarrow-\infty} f(x)=\lim _{x \rightarrow \infty} f(x)=6\), it is known that the function approaches the value 6 as \(x\) tends towards infinity or negative infinity. This implies that the function has a horizontal asymptote at \(y=6\) as the value of \(x\) tends either towards positive infinity or negative infinity.

Step 3: Sketch the Graph

Using the above informations, start by drawing a sketch of the horizontal asymptote at \(y=6\). Coming to the critical point at \(x=2\), make a point below \(y=6\) since it's a minimum. The left part of the graph should be decreasing towards the asymptote and the right part should be increasing towards the asymptote.

Key Concepts

Critical Number

A critical number is a point on the graph of a function where the derivative is zero or undefined. In simpler terms, it is where the slope of the tangent line to the graph is horizontal, meaning the graph either has a peak, a trough, or a horizontal inflection point. In our exercise, the critical number is given as \(x=2\). This means that at \(x=2\), the slope of the function \(f\) is zero, and the nature of the function changes. Before \(x=2\), the function is decreasing, and after \(x=2\), it starts increasing. Therefore, this critical number at \(x=2\) represents a minimum since the function changes from decreasing to increasing. Understanding critical numbers is vital as they help identify where key changes in the behavior of the function occur.

Derivative

The derivative of a function represents the rate of change or the slope of the function at any given point. In mathematical terms, it is the function \(f'\) that gives us this information. For our given exercise, the derivative helps determine where the function \(f\) decreases or increases.

• When \(f'(x)<0\), the function is decreasing because the slope is negative.
• When \(f'(x)>0\), the function is increasing since the slope is positive.

These insights let us conclude there is a turning point at the critical number \(x=2\) where the function transitions from decreasing to increasing. The derivative acts as a tool to shape our understanding of the overall layout of the graph.

Horizontal Asymptote

Horizontal asymptotes are horizontal lines that a graph of a function approaches as \(x\) goes to positive or negative infinity. They indicate the end behavior of a function. In this exercise, the condition \(\lim_{x \to -\infty} f(x) = \lim_{x \to \infty} f(x) = 6\) informs us that the values of \(f(x)\) approach 6 as \(x\) heads towards either infinity.
This symmetric behavior suggests that the function \(f\) is "flattening out" or stabilizing around \(y=6\) at the extreme ends of the graph. The existence of a horizontal asymptote helps elucidate how the function behaves in the long run.

Function Behavior

Function behavior refers to how a function acts across its domain, mainly focusing on where it increases, decreases, or stabilizes. In our scenario:

• For \(x<2\), the function is decreasing because \(f'(x)<0\). This indicates a downward slope.
• At \(x=2\), there’s a critical point indicating it’s the minimum due to the transition from a negative to a positive derivative.
• For \(x>2\), the function is increasing with \(f'(x)>0\), which means there's an upward slope.

Under the influence of the horizontal asymptote \(y=6\), the function stabilizes towards this asymptote as \(x\) tends to \(\pm\infty\). The function is structured as a concave smile that enforces the existence of a minimum point at the critical number, with its arms ever reaching outward toward the horizontal asymptote.

Graph Sketching

Graph sketching is an art form of conveying the behavior and feature of a function visually. It involves using derivative information to draw conclusions about the shape and direction of the graph. In this exercise, you use the details from the previous steps to draw:

• A horizontal line at \(y=6\) representing the asymptote.
• A minimum point at \(x=2\), slightly below the asymptote line, since it’s a local minimum.
• The left side of the graph sloping downward towards the minimum, showing a decreasing behavior.
• The right side of the graph sloping upward past the minimum, indicating an increase in function value.

This visual arrangement, guided by derivative clues and asymptotic behavior, makes depicting the graph an intuitive and insightful process.