Question

Sketch a graph of a function \(f\) having the given characteristics. (There are many correct answers.) $$ \begin{aligned} &f(-1)=f(3)=0\\\ &f^{\prime}(1) \text { is undefined. }\\\ &f^{\prime}(x)<0 \text { if } x<1\\\ &f^{\prime}(x)>0 \text { if } x>1\\\ &f^{\prime \prime}(x)<0, x \neq 1\\\ &\lim _{x \rightarrow \infty} f(x)=4 \end{aligned} $$

Short answer

By considering these aspects, you should be able to sketch a graph that meets all the given conditions. Remember, there can be multiple correct answers because the exercise has not specified a unique function, but rather a set of characteristics that the function's graph must possess.

Step-by-step solution

Function values

Start by plotting the given points. We know that \(f(-1)=f(3)=0\), so we place points at \((-1,0)\) and \((3,0)\).

Slope of function

Next, interpret the characteristics of \(f'(x)\). We are told that \(f'(x)<0\) if \(x<1\) and \(f'(x)>0\) if \(x>1\). This informs us that the function is decreasing for \(x<1\) and increasing for \(x>1\). We also know that \(f'(1)\) is undefined, implying there's a corner or cusp at \(x=1\). Draw the graph in such a way that these criteria are met.

Concavity of function

We are also told that \(f''(x)<0\) for \(x \neq 1\). This tells us that the function is always concave down except at \(x=1\). Draw the curve such that it is concave down on its entire length except at \(x=1\), where the concavity might change.

Limits

Finally, note the given limit, \(\lim _{x \rightarrow \infty} f(x)=4\). This informs us that as \(x\) approaches infinity, the value of the function approaches 4. Make sure your graph 'levels off' towards the value \(y=4\) as \(x\) goes to positive infinity.

Key concepts

Limits of Functions

Understanding the limits of functions is a fundamental concept in calculus, helping us grasp the behavior of a function as the input approaches a certain value. In the given exercise, we encounter the limit \(\lim _{x \rightarrow \infty} f(x)=4\), which tells us that as \(x\) becomes very large, \(f(x)\) gets closer and closer to 4.

This concept is crucial when sketching graphs as it sets an 'asymptotic behavior'—a boundary condition for the function at infinity. Our function won't cross this line as \(x\) grows beyond bounds, so the graph should show a horizontal approach towards the line \(y=4\) as \(x\) extends to the right.

Function Concavity

The concavity of a function refers to the direction in which a curve bends. In our function, we have \(f''(x)<0\) for \(x eq 1\), which means the graph is concave down everywhere except possibly at \(x=1\).

Visualizing Concavity

Think of concave down as resembling the shape of an upside-down bowl. When sketching, this indicates that the function curves downward, and at any point (except \(x=1\)), a line tangent to the curve will lie above the curve itself.

This behavior is essential for understanding the way a function will bend as you move along the graph and is directly influenced by the function's second derivative, \(f''(x)\).

Undefined Derivatives

An undefined derivative, such as \(f'(1)\) in our exercise, often points to a sharp turn, cusp, or vertical tangent in the graph. These occurrences break the smoothness of the curve.

Interpreting Discontinuities

When you come across an undefined derivative, it's important to depict this on the graph correctly. In this instance, be sure to illustrate a clear, distinct change in direction at \(x=1\) without rounding out the corner.

It's helpful to think that the slope of the tangent line at this point does not exist, and it's this absence of a clear slope that makes our derivative undefined.

Increasing and Decreasing Functions

The concept of increasing and decreasing functions relates to how the output of the function (the \(y\)-value) changes as the input (the \(x\)-value) increases. From the exercise, we know that \(f'(x)<0\) if \(x<1\) meaning the function is decreasing; as \(x\) grows, \(f(x)\) falls. Conversely, \(f'(x)>0\) if \(x>1\) implies the function is increasing; as \(x\) increases, so does \(f(x)\).

Drawing it Out

When sketching the graph, this tells us to slope downwards to the left of \(x=1\) and slope upwards to the right.

Remember that the first derivative test can be used to find where a function is increasing or decreasing by analyzing the signs of the derivative. This aspect informs not just the slope but also the nature of local maxima and minima, which are vital in sketching the overall shape of the graph.