Problem 54 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

First identify \(x=2\) as the critical number and recognize that it represents a local minimum due to the sign changes in the derivatives. Then acknowledge the information given by the limits at infinity to realize that the function approaches \(6\) both as \(x\) trends towards negative and positive infinity. Utilize all this information to sketch a graph that fulfills the given conditions.

Step 1: Identify critical number

The critical number is given as \(x=2\). This value of \(x\) is the point at which the derivative \(f'(x)\) changes sign. This will typically identify a minimum or maximum value of the function \(f\).

Step 2: Consider the sign of the derivative

According to the exercise, \(f'(x)<0\) for \(x<2\) and \(f'(x)>0\) for \(x>2\). This indicates that \(f\) is decreasing to the left of \(x=2\) and increasing to the right of \(x=2\). Therefore, \(x=2\) must be a local minimum of the function \(f\).

Step 3: Consider the limits at infinity

We're also given that \(\lim_{x \rightarrow -\infty} f(x) = 6\) and \(\lim_{x \rightarrow \infty} f(x) = 6\). This suggests that the function approaches the value of \(6\) both as \(x\) goes to negative infinity and as \(x\) goes to positive infinity.

Step 4: Sketch the graph

Put all the information together to sketch the graph. A possible function that fits the description would have the following features: - Decreasing to the left of \(x=2\) and increasing to the right, indicating \(x=2\) as a local minimum. - The function heads towards the value \(6\) as \(x\) goes to negative and positive infinity. - Since there's no information about what value \(f\) takes at \(x=2\), and it's a local minimum, an arbitrary value can be chosen. For simplicity, one could also let \(f(2) = 6\). Combining these facts, a graph can be sketched.

Key Concepts

Critical Number

In calculus, a critical number is a value of x where the derivative of a function, denoted as f'(x), is either zero or undefined. It's an essential concept because it helps locate potential peaks (maxima), valleys (minima), or points of inflection in the graph of a function.

The problem at hand specifies that x=2 is the function's only critical number. This means that at x=2, the behavior of the function changes, providing us a clue that there might be either a peak or a valley at this point, which we'll explore further.

Derivative of a Function

The derivative of a function represents the rate at which the function's value is changing at any given point. Think of it like the slope of the tangent line to the function's graph at any point x. For a function that is continuously differentiable, looking at the sign of its derivative is a handy way to determine the function's increasing or decreasing behavior.

If f'(x) > 0, the function is increasing, and if f'(x) < 0, it’s decreasing. By understanding the derivative, we can predict the overall shape of the function's graph beyond just the critical points.

Limits at Infinity

When we investigate the limits at infinity of a function, we are studying the behavior of the function as x approaches infinitely large positive or negative values. It's about understanding the end behavior of a function's graph.

In the given problem, we have both \( \lim_{{x \rightarrow -\infty}} f(x) = 6 \) and \( \lim_{{x \rightarrow \infty}} f(x) = 6 \), indicating that as x becomes very large or very small, the function levels off towards the horizontal asymptote at y=6. This tells us that the graph has a 'ceiling' and 'floor' to which it tries to settle in either direction.

Local Minimum

A local minimum refers to a point on the graph of a function where its value is lower than all nearby points. It's a dip or a trough in the graph, a point of relative low height. Specifically, for our problem, since the derivative f'(x) changes from negative to positive at x=2, it indicates a transition from decreasing to increasing behavior, identifying x=2 as the location of a local minimum.

Analyzing a local minimum is crucial to understanding the function's local behavior and can tell us a lot about the function's overall shape and possible optimal points.