Problem 26 - University Calculus: Early Transcendentals

Question

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$F(x)=-\frac{1}{x},-2 \leq x \leq-1$$

Step-by-Step Solution

Answer

Maximum is 1 at \(x = -1\); Minimum is \(1/2\) at \(x = -2\).

Step 1: Identify the function and interval

The given function is \( F(x) = -\frac{1}{x} \), and the interval is \([-2, -1]\). We need to find the absolute maximum and minimum values of this function on this interval.

Step 2: Find critical points

To find the critical points, we first need to find the derivative of \( F(x) \). Calculate \( F'(x) \) by differentiating \( F(x) = -\frac{1}{x} \). The derivative is \( F'(x) = \frac{1}{x^2} \). Critical points occur where \( F'(x) = 0 \) or is undefined. \( F'(x) \) is never zero, but it is undefined at \( x = 0 \), which is not within our interval, so there are no critical points within \([-2, -1]\).

Step 3: Evaluate function at endpoints

Since there are no critical points within the interval, evaluate \( F(x) \) at the endpoints \( x = -2 \) and \( x = -1 \). Calculate:- \( F(-2) = -\frac{1}{-2} = \frac{1}{2} \),- \( F(-1) = -\frac{1}{-1} = 1 \).

Step 4: Determine absolute extrema

Compare the values obtained at the endpoints to find the absolute extrema:- The absolute maximum value is \( 1 \) at \( x = -1 \).- The absolute minimum value is \( \frac{1}{2} \) at \( x = -2 \).

Step 5: Graph the function and identify extrema points

To graph the function, plot \( F(x) = -\frac{1}{x} \) over the interval \([-2, -1]\). Mark the points where the extrema occur:- Point \((-1, 1)\) for the maximum.- Point \((-2, \frac{1}{2})\) for the minimum.

Key Concepts

Absolute Extrema

Absolute extrema refer to the highest and lowest values that a function can reach on a specific interval. These are known as absolute maximum and absolute minimum. To find these values, we generally look at critical points and the function's endpoints within the interval. The absolute extrema are essential because they tell us which values the function cannot exceed or dip below within the given range.

• First, identify the function and interval.
• Then, locate the critical points, as these are potential candidates for absolute extrema.
• Finally, compare function values at critical points and endpoints to find the absolute max and min.

In our scenario, no critical points were found within the interval, hence the function values only at the endpoints were used.

Critical Points

Critical points of a function are points where the derivative is either zero or undefined. At these points, a function might change direction, indicating potential extrema—either relative or absolute.
To find critical points, follow these steps:

• Take the derivative of the function.
• Set the derivative equal to zero and solve for x.
• Identify where the derivative is undefined as these are also critical points.

In our example, the derivative of the function was never zero. However, it was noted to be undefined at a point outside the interval we considered, so there were no critical points on extit{the interval} [-2, -1]. Thus, the critical points did not contribute to finding the absolute extrema in this case.

Derivative

A derivative represents the rate at which a function is changing at any point on its graph. It is the slope of the tangent line at any point on the curve. Calculating the derivative is essential to understanding where critical points might occur because they often correspond to points of zero slope or undefined slope.
Here's how you calculate it:

• For the function \( F(x) = -\frac{1}{x} \), apply differentiation rules to find the derivative.
• The derivative, \( F'(x) = \frac{1}{x^2} \), is positive as \( x^2 \) is always positive over real numbers.
• This derivative doesn’t equal zero within the interval but identified the point where it's undefined.

Understanding derivatives is critical in finding extrema as they demonstrate how the function behaves locally.

Graphing Functions

Graphing is a powerful tool in visualizing and understanding a function's behavior. It helps identify where a function may reach its maximum or minimum values. A graph can provide intuition beyond pure computation.
When graphing a function like \( F(x) = -\frac{1}{x} \), follow these steps:

• Identify and plot key points such as endpoints, and extrema.
• Note the behavior approaching vertical and horizontal asymptotes.
• Understand that graphing over an interval means highlighting only the section of the function in that range.

Through graphing, we observed the highest value at \((-1, 1)\) and the lowest at \((-2, \frac{1}{2})\), confirming the analytical results and offering a clear picture of how the function behaves on the specified interval. By approaching graphing in steps, it enhances the understanding and interpretation of the function’s characteristics.