Question

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=\frac{4}{1+x^{2}} $$

Short answer

Relative extrema and inflection points can be found by deriving the original equation and setting it equal to zero, then evaluating the original function at these points. The exact points are determined by the results.

Step-by-step solution

Graphing the function

Plot the function \(f(x)=\frac{4}{1+x^{2}}\) in a graphing utility. The graph will give a basic idea of the shape of the function.

Understanding the Expected Results

An extremum point of a function is any point where the function changes from increasing to decreasing (meaning a relative maximum) or from decreasing to increasing (meaning a relative minimum). On the other hand, an inflection point is a point on the curve of the graph where the curve changes the direction of its curvature, i.e. from being concave upward to concave downward, or from concave downward to concave upward.

Finding the Relative Extrema

The relative extrema of a function can be found by setting the derivative of the function equal to zero, finding the 'x' values, then evaluating the original function at these 'x' values. Here, you find the first derivative \(f'(x)\), set it equal to zero and solve for x. The second derivative \(f''(x)\) will give information about the concavity of the function.

Finding the Inflection Points

The inflection points can be found by determining where the function changes concavity, by setting the second derivative equal to zero, solving for 'x', and then evaluating these 'x' values in the original function.

Key concepts

Relative Extrema

When we talk about relative extrema, we refer to the high and low points of a function within a certain interval. Think of them as the peaks and valleys within a specific range on a graph. To determine these points, we take the derivative of the function and look for where it equals zero. These are the critical points. But not all critical points are relative extrema. To identify an extremum, we check the intervals around the critical points to see if the function changes from increasing to decreasing (indicating a relative maximum) or vice versa (indicating a relative minimum).

For our function, the derivative is found and set to zero. The calculated 'x' values that result are potential locations of relative extrema which must be further examined with the second derivative or by analyzing the sign change of the first derivative around those points.

Points of Inflection

Points of inflection are the locations on a graph where the curvature changes direction. In other words, these are the points where a function's graph goes from curving upwards to curving downwards, or vice versa. To find points of inflection, we use the second derivative of the function. By setting the second derivative equal to zero and solving for 'x', we find potential points of inflection. However, we must verify a sign change in the concavity by checking the intervals around these 'x' values.

For the function \( f(x)=\frac{4}{1+x^{2}} \), it’s essential to compute the second derivative, set it equal to zero, and determine where the concavity changes from concave up to concave down or vice versa to confirm the points of inflection.

Derivative of a Function

The derivative of a function is a foundational concept in calculus, representing the rate of change of the function or the slope of the tangent line at any given point. It describes how the output of the function changes as the input changes. To find the derivative, we use rules of differentiation, such as the power rule, product rule, and chain rule, depending on the form of the function.

The direction in which a function is increasing or decreasing and its rate of change at any given point is given by the sign and magnitude of the first derivative, respectively. When graphing \( f(x)=\frac{4}{1+x^{2}} \) and calculating its derivative, we are given significant insight into the behavior of the function across different values of 'x'.

Concavity of a Graph

Concavity of a graph refers to how a curve bends. A graph can be concave up (like a cup facing upwards) or concave down (like a cup facing downwards). The second derivative of a function gives us information about the concavity at different points. If the second derivative is positive for a range of 'x' values, the graph is concave up in that interval. If it is negative, the graph is concave down.

By examining the second derivative of \( f(x)=\frac{4}{1+x^{2}} \), we can determine where the function's graph curves upward or downward. Analyzing the concavity helps us not just in identifying points of inflection but also in getting a better grasp of the overall geometry and shape of the function's graph as it relates to various real-world applications.