Problem 12 - Applied Calculus

Question

Sketch the graph of a function on the inter val \(0 \leq x \leq 10\) with the given properties. Has global maximum at \(x=0\), global minimum at \(x=10\), and no other local maxima or minima.

Step-by-Step Solution

Answer

Decrease continually from maximum at \(x=0\) to minimum at \(x=10\).

Step 1: Understanding the Function's Behavior

We are given a function on the interval \(0 \leq x \leq 10\). The function has a global maximum at \(x=0\) and a global minimum at \(x=10\). No other local maxima or minima exist within the interval. This information implies that the function must decrease continuously from \(x=0\) to \(x=10\).

Step 2: Determining the Start and End Points

Begin by plotting the point \((0, y_0)\), where \(y_0\) is the maximum value of the function, and the point \((10, y_{10})\), where \(y_{10}\) is the minimum value of the function. This establishes the endpoints of the function's graph: the starting point at \(x=0\) should be the highest, and the endpoint at \(x=10\) should be the lowest.

Step 3: Sketching the Continuous Descent

Draw a smooth curve starting at the point \((0, y_0)\) and ending at the point \((10, y_{10})\) that continuously decreases without any peaks or valleys between these points. This curve should reflect the continuous descent required by the given properties. Keep in mind that the path of the curve can vary (straight line, exerting, etc.) as long as it meets the decreasing manner specified.

Step 4: Verification of Graph's Features

Review the sketched graph to ensure that it adheres to the problem's conditions: a global maximum at \(x=0\), a global minimum at \(x=10\), and no other local extrema within the interval. Make adjustments if needed to ensure the continuous descending property is correctly represented.

Key Concepts

Global Maximum

The term "global maximum" refers to the highest point over the entire domain for a particular function. In the exercise, we see that the global maximum occurs at the beginning of the interval, at \(x=0\). At this point, the value of the function, denoted as \(y_0\), is greater than or equal to any other value of the function within the interval \(0 \leq x \leq 10\).
Understanding global maximum is crucial because it defines the peak of the graph.

• It is the point where the function reaches its highest value.
• In our graph, the curve starts at this maximum value \((0, y_0)\).

Ensure that when you sketch graphs, the location and value of this point are clearly indicated to meet the function's specified maximum conditions.

Global Minimum

Similarly, the "global minimum" signifies the lowest point in the function's entire range. For our function, this minimum is found at the end of the interval, \(x=10\).
The function's value at this point is denoted as \(y_{10}\), and it represents the smallest value that the function takes on the interval \(0 \leq x \leq 10\).
Grasping the concept of the global minimum allows us to understand where the graph hits its lowest point.

• It's the lowest value that the function touches throughout its domain.
• In the sketch, the curve should end here, highlighting the descent to \((10, y_{10})\).

By clearly identifying the global minimum, you ensure the graph reflects the function's actual behavior perfectly.

Continuous Function

A "continuous function" is one that is smooth throughout its domain, without breaks, jumps, or holes. Taking this property into account is essential when sketching the function’s graph. In our exercise, the function is continuous over the interval \(0 \leq x \leq 10\).
This means that from the global maximum at \(x=0\) to the global minimum at \(x=10\), the function's graph must be a seamless, uninterrupted line or curve.
The continuity of the function is reflected in how smoothly the descent is drawn between these two points.

• The graph must exhibit a continuous decrease - no sudden changes in direction.
• This smooth descent ensures the graph adheres to the mathematical definition of continuity.

By maintaining a continuous line or curve, you make sure that your graph accurately represents the behavior described in the exercise.