Question

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ g(x)=4\left(1+\frac{1}{x}+\frac{1}{x^{2}}\right), \quad[-4,5] $$

Short answer

The absolute maximum of the function on the interval [-4,5] is 4.88 at x = 5, and the absolute minimum is 3 at x = -4.

Step-by-step solution

Identify the critical points

The critical points of a function are the points where the function's derivative is either zero or undefined. First, find the derivative of the function \( g(x) \). So, \( g'(x) = -4 \cdot \left(\frac{1}{x^2} + \frac{2}{x^3}\right) \). Set this equal to 0 to solve for \( x \) to find the critical points. This yields \( x = -1 \). However, \( x = -1 \) is not in the interval [-4,5]. Thus there are no critical points in the interval.

Evaluate the function at the endpoints of the interval

Evaluate the function at \( x = -4 \) and \( x = 5 \). So, \( g(-4) = 4 \cdot \left(1 - \frac{1}{4} + \frac{1}{16}\right) = 3 \) and \( g(5) = 4 \cdot \left(1 + \frac{1}{5} + \frac{1}{25}\right) = 4.88 \).

Determine the absolute maxima and minima

The absolute maximum of \( g(x) \) on the interval [-4,5] is the largest value, which is \( g(5) = 4.88 \), and the absolute minimum is the smallest value, which is \( g(-4) = 3 \).

Key concepts

Critical Points

Understanding critical points is fundamental in calculus for identifying where a function may have a local maximum or minimum. These points occur where the derivative of a function equals zero or where the derivative does not exist. In the context of finding absolute extrema on a closed interval, critical points can indicate potential locations of these extrema within the interval.

When approaching an exercise involving critical points, it’s essential to find the derivative of the function first and then solve for when this derivative is zero. However, it’s crucial to remember that critical points found must lie within the interval in question. If they fall outside the interval, as with the point x = -1 in our example, they do not influence the outcome. The understanding of critical points helps in pinpointing exact locations on a graph where interesting behavior occurs - such as peaks or troughs - which are valuable for analyzing the function's behavior.

Derivative of a Function

The derivative of a function is a cornerstone in calculus representing the rate of change or the slope of the function at any given point. For a function g(x), the derivative is denoted g'(x).

When looking for the extremes of a function, calculating the derivative and setting it equal to zero allows you to find where the function’s slope changes direction – a potential extremum. It's worth noting that not every critical point will result in an extremum, but every extremum must correspond with a critical point or boundary of the interval. When using the derivative, ensure that calculations are made accurately, as even a small error can lead to incorrect conclusions about the function’s behavior.

Closed Interval

A closed interval is a range of numbers that includes both of its endpoints. In notation, it's written as [a,b], meaning all numbers from 'a' up to 'b' inclusive are part of the interval. This is vital when finding absolute extrema, as it informs us that we must consider the function's values at these endpoints as potential candidates for maximums or minimums.

Evaluating Endpoints

In our exercise, evaluating the function at the endpoints of the closed interval is crucial. In the interval [-4,5], both -4 and 5 are part of the analysis. As functions can behave differently at their endpoints, ignoring these values could result in missing out on the true absolute extrema. Hence, always ensure to substitute the endpoints into the function as part of your procedure.

Graphing Utility

A graphing utility is an indispensable tool in visualizing the behavior of mathematical functions. It translates complex algebraic expressions into a visual graph that can be used to verify solutions such as absolute extrema found algebraically. When using graphing utilities in exercises, it's important to input the function correctly and set the appropriate viewing window to observe the interval of interest.

Confirming Results

In the textbook exercise, the usage of a graphing utility is recommended to confirm the absolute extrema identified by the analytical method. This not only provides visual confirmation but also helps identify any potential errors made during the calculation process. Especially for students, integrating a graphing utility into their learning can greatly enhance comprehension of the function's overall behavior.