Let's analyze and further explore the steps for solving this problem based on the image provided.

The task is to analyze the function:

$y = \frac{5x}{x^2 + 9}$

1. Domain

The function's domain, as given in the image, is $(- \infty, \infty)$, which is correct because the function has no points of discontinuity (i.e., there are no values of $x$ for which $x^2 + 9 = 0$).

2. Symmetry

The function was correctly identified as odd and symmetric about the origin. This conclusion comes from testing $f(-x) = -f(x)$, which holds for this function.

3. First Derivative ($y'$)

The image shows the initial setup to find $y'$:

$y' = \frac{d}{dx}\left(\frac{5x}{x^2 + 9}\right)$

To find the derivative of this function, we use the quotient rule, which states that if $y = \frac{u}{v}$, then

$y' = \frac{v u' - u v'}{v^2}$

For $y = \frac{5x}{x^2 + 9}$:

• $u = 5x$ and $v = x^2 + 9$
• $u' = 5$
• $v' = 2x$

Applying the quotient rule:

$y' = \frac{(x^2 + 9)(5) - (5x)(2x)}{(x^2 + 9)^2}$

Simplifying each term:

$y' = \frac{5x^2 + 45 - 10x^2}{(x^2 + 9)^2} = \frac{-5x^2 + 45}{(x^2 + 9)^2} = \frac{5(9 - x^2)}{(x^2 + 9)^2}$

Further Analysis Needed

To complete the analysis of this function, the following steps are needed:

1. Identify Critical Points: Set $y' = 0$ to find where the function has horizontal tangents.
2. Determine Concavity and Points of Inflection: Find the second derivative $y''$, analyze its sign to determine intervals of concavity, and find inflection points.
3. Find Asymptotes: Analyze the behavior of $y$ as $x \to \pm \infty$.
4. Plot Key Points and Extreme Points: Evaluate the function at critical points, intercepts, and asymptotes, if any, to sketch the graph.

Would you like details on finding critical points, concavity, asymptotes, or another part of the analysis?

Related Questions

1. What are the steps for finding the critical points of this function?
2. How do you determine concavity and identify points of inflection?
3. What asymptotes does the function $y = \frac{5x}{x^2 + 9}$ have?
4. How do you confirm whether a function is odd or even?
5. Why is the domain of this function all real numbers?

Tip

When analyzing functions, remember that symmetry (odd/even) and asymptotic behavior can simplify the graphing process significantly, especially when verifying critical points and concavity.