Question

Sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. \(f\) is continuous, but not necessarily differentiable, has domain \([0,6]\), and has one local minimum and one local maximum on \((0,6)\).

Short answer

A continuous function can be graphed with a local minimum at \(x = 2\) and a local maximum at \(x = 4\) within \([0,6]\).

Step-by-step solution

Understand the Properties

We need to sketch a continuous function that spans from \(x = 0\) to \(x = 6\) and has one local minimum and one local maximum in the open interval \((0,6)\). The function should not have any jumps or breaks, as it is continuous.

Determine Critical Points

Identify positions for the critical points. Since the local minimum and maximum must be within the interval \((0,6)\), choose two points such as \(x = 2\) and \(x = 4\) where the local minimum and local maximum will occur. This ensures these extrema aren't at the endpoints.

Sketch Intervals and Behavior

From \(x = 0\) to \(x = 2\), the function should decrease to the local minimum at \(x = 2\). From \(x = 2\) to \(x = 4\), the function increases to the local maximum at \(x = 4\). Finally, from \(x = 4\) to \(x = 6\), it should decrease again. This pattern ensures the presence of exactly one local minimum and maximum.

Ensure Continuity

Ensure the function is continuous on the closed interval \([0,6]\). It should have no abrupt jumps or breaks between \(x = 0\) and \(x = 6\), smoothly connecting all segments of the curve.

Check Domain Coverage

Verify that the function covers the entire domain \([0,6]\). The sketch should clearly begin at \(x = 0\) and end at \(x = 6\), without leaving any gaps in the domain.

Key concepts

Local Minimum

A local minimum in a function is a point where the function value is lower than at all nearby points. Think of it as a small valley on a graph. To find a local minimum, imagine a small region around a point; if the function dips below this area and then rises back up, you've spotted a potential local minimum.
When sketching, observe if the function's slope changes from negative to positive at a point. This is an indicator of a local minimum. For the given exercise, we chose a local minimum at around \(x = 2\). This means the function decreases until it hits \(x = 2\), reaching its lowest point locally, and then starts to increase as we move further along the graph. To enforce continuity, ensure the transition into and out of this local minimum is smooth, with no sharp corners or breaks.

Local Maximum

A local maximum is the opposite of a local minimum. It's where the function reaches a peak in a small region, like a hilltop you're standing on. A local maximum will occur when the slope of the function changes from positive to negative. This pattern helps in identifying such a point.
In the exercise solution, we positioned the local maximum at \(x = 4\). Here, the function climbs up to this point and then begins its descent. This ensures a peak without reaching the absolute domain endpoints. It's important that this transition to and from the local maximum is gradual and without any harsh changes to maintain the function's continuous nature. Well-placed local maxima are essential in sketching the graph as they direct the function's flow while emphasizing dynamic changes in behavior.

Critical Points

Critical points are crucial in understanding a function's behavior since they highlight where the function's direction shifts. These are typically places where the derivative is zero or undefined, signifying changes in the slope.
Locating critical points in the exercise involved choosing strategic values within the interval \(0,6\). The chosen critical points occurred at \(x = 2\) and \(x = 4\), marking where the local minimum and maximum sit on the graph. Identifying these points helps in predicting where the function bends, ensuring the function addresses the specified local minimum and maximum.
By examining changes in slope around these critical points, you can infer how to sketch the function's rise and fall over the interval. Accurately determining these points aids not only in sketching but also in understanding the nuances of the function's movement and potential.

Graph Sketching

Graph sketching is the art of visually interpreting the function's behavior based on known characteristics. This skill involves plotting critical transitions and ensuring continuity across the domain.
From the exercise, we began sketching by planning how the function behaves from each critical point. Starting at the interval \(x = 0\) towards the local minimum at \(x = 2\), and then rising to the local maximum at \(x = 4\), before finally coming down again to \(x = 6\). This structured journey ensures that all required properties—a local minimum, a local maximum, and the function's continuity from \(x=0\) to \(x=6\)—are showcased.
Good graph sketching respects the function's continuity, creating a smooth, flowing curve without jumps. By thoroughly understanding each segment's motion between critical points, you can construct a meaningful and accurate depiction of the function's lifespan over the given interval.