Graph each function, considering the domain, critical...

Graph each function, considering the domain, critical points, symmetry, relative extrema, regions where the function is increasing or decreasing, inflection points, regions where the function is concave upward or concave downward, intercepts where possible, and asymptotes where applicable. (Hint: In Exercise 21, use the result of Exercise 1. In Exercises 25-27, recall from Exercise 68 in Section 3.1 on Limits that $\lim _{x \rightarrow \infty} x^{n} e^{-x}=0 . )$ $$f(x)=2 x+\frac{10}{x}$$

Key Concepts

Domain Analysis

This concept involves determining the set of all x-values for which the function is defined. It requires identifying restrictions such as division by zero or taking even roots of negative numbers. Ensuring the domain is correctly established is crucial because it defines the context for all subsequent analysis of the function’s behavior.

First Derivative and Critical Points

The first derivative of a function provides information about the rate of change or the slope at each point on its graph. By setting the derivative equal to zero or identifying where it does not exist, one can find the critical points. These points are candidates for local extrema (maximums or minimums) and play a significant role in understanding where the function is increasing or decreasing.

Second Derivative and Inflection Points

The second derivative gives insight into the curvature or concavity of a function. Evaluating the sign of the second derivative reveals intervals where the function is concave upward (curving up) or concave downward (curving down). Points where the second derivative changes sign are known as inflection points, marking changes in the curvature direction of the function.

Intercept Determination

Intercepts, specifically x-intercepts and y-intercepts, are found by setting the function equal to zero for x-intercepts and evaluating the function at x = 0 for the y-intercept. Identifying these points is essential for accurately plotting the graph of the function as they provide fixed reference points where the graph crosses the coordinate axes.

Asymptotic Behavior

The analysis of asymptotes involves determining if the function approaches a certain line as x tends to infinity, negative infinity, or other critical points. Vertical asymptotes occur where the function increases or decreases without bound near a specific x-value, while horizontal or oblique asymptotes describe the behavior of the function as x approaches infinity. Limits are often used to rigorously establish these asymptotic behaviors.

Symmetry Analysis

Symmetry analysis assesses whether a function is even, odd, or neither by substituting -x for x in the function. This property is useful because even functions exhibit symmetry with respect to the y-axis and odd functions with respect to the origin. Recognizing symmetry can simplify the graphing process and provide insights into the general behavior of the function.

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