Problem 21 - Essential Calculus Early Transcendentals

Question

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=1-\sqrt{x}$$

Step-by-Step Solution

Answer

The absolute maximum is 1 at x=0, and there is no local or absolute minimum in the domain of non-negative x.

Step 1: Understanding the Function

The function given is \( f(x) = 1 - \sqrt{x} \). This is a transformation of the basic square root function \( \sqrt{x} \). We need to understand the impact of each component. The \( -\sqrt{x} \) flips the graph of \( \sqrt{x} \) vertically, and the \(+1\) translates the graph upwards by 1 unit.

Step 2: Sketching the Function

Start by sketching the basic square root function \( \sqrt{x} \), which starts at the origin (0,0) and extends up and to the right. For \( -\sqrt{x} \), reflect this curve across the x-axis, so it starts at the origin and extends down and to the right. Finally, shift the entire graph up by 1 unit to achieve \( f(x) = 1 - \sqrt{x} \). This graph starts at (0,1) and decreases as x increases.

Step 3: Finding the Absolute Maximum

Looking at the graph, the maximum value of \( f(x) = 1-\sqrt{x} \) is at its starting point because for all positive x, the graph decreases. Therefore, the absolute maximum occurs at x = 0, where \( f(0) = 1 \).

Step 4: Finding the Absolute Minimum

As x approaches infinity, \( -\sqrt{x} \) approaches negative infinity. Therefore, as x increases, \( f(x) = 1 - \sqrt{x} \) continues to decrease towards negative infinity. However, for a real number domain where x is non-negative, there is no absolute minimum since the function continues to decrease indefinitely.

Step 5: Conclusion on Local Minimum and Maximum

Since \( f(x) = 1 - \sqrt{x} \) is a strictly decreasing function from x = 0 to infinity, it does not have any local minimum or maximum within this domain beyond the absolute maximum found at x = 0.

Key Concepts

Absolute Maximum

When we talk about absolute maximum, we mean the highest point in the range of a function on its domain. In our case, the function is \( f(x) = 1 - \sqrt{x} \). To understand this, let's explore how the graph behaves. This graph starts at the point \( (0,1) \) and continuously decreases as \( x \) increases. An important point to note here is that the absolute maximum is the greatest value that the function achieves. For this specific function, the absolute maximum is observed right at the point of origin on the graph, \( x = 0 \), where \( f(0) = 1 \). Since there is no higher value that \( f(x) \) can take on within its valid domain of \( x \geq 0 \), \( f(0) = 1 \) is the absolute maximum.

• The topmost point in the function's range.
• Occurs at the beginning of the function's domain.
• The highest value the function can produce.

Local Minimum and Maximum

Local minimum and maximum refer to the points in the graph where a function changes direction from increasing to decreasing or decreasing to increasing, respectively. However, it's important to understand that not all functions have these points. In the function \( f(x) = 1 - \sqrt{x} \), the shape of the graph shows that it is strictly decreasing, meaning it does not turn back upwards or change its direction at any point. This continual decrease implies there are no local maxima or minima within the domain.

• Local extrema are points where increments stop and decrements start, or vice versa.
• If the function only decreases or increases, local extrema won't appear.
• In this case, since \( f(x) \) monotonically decreases, there are neither local minimums nor maximums.

Function Transformations

Function transformations involve changes that modify the graph of a basic function in various ways without altering its overall shape or direction. With \( f(x) = 1 - \sqrt{x} \), we see a combination of such transformations from the basic \( \sqrt{x} \), or square root function. There are two main transformations applied here:- **Vertical Reflection**: The negative sign before the square root, \( -\sqrt{x} \), reflects the graph across the x-axis. This turns the usual upwards opening root graph downwards.- **Vertical Shift**: The addition of 1, shown as \( 1 - \sqrt{x} \), moves the entire graph upwards by 1 unit. To visualize these transformations:

• Start with the basic \( \sqrt{x} \) graph, originating at \( (0,0) \).
• Reflect it vertically to point downwards, resulting in \( -\sqrt{x} \).
• Shift every point of this graph upwards by one unit.

These transformations help us understand key characteristics and behavior of \( f(x) \), aiding in easier graph sketching and extrema identification.