The provided image contains a mathematical function and a couple of questions related to its analysis. Here's what the image shows:

Function:

$f(x) = \frac{2x}{x^2 + 1}$

Questions:

1. Positividad y negatividad (Positivity and negativity)
2. Crecimiento y decrecimiento. ¿Extremos? (Growth and decrease. Extrema?)

Let’s proceed with a detailed step-by-step analysis.

Part (a): Positivity and Negativity of $f(x)$

• Positivity: $f(x) > 0$ $\frac{2x}{x^2 + 1} > 0$ The numerator is $2x$, and the denominator $x^2 + 1$ is always positive for all real $x$. Thus, the sign of the function depends entirely on the sign of the numerator.

  • $2x > 0 \quad \Longrightarrow \quad x > 0$

  Hence, the function is positive for: $x > 0$

• Negativity: $f(x) < 0$

  • $2x < 0 \quad \Longrightarrow \quad x < 0$

  Hence, the function is negative for: $x < 0$

• At $x = 0$: $f(0) = \frac{2(0)}{0^2 + 1} = 0$ Thus, the function is zero at $x = 0$.

Part (b): Growth, Decrease, and Extrema

To determine intervals of increase and decrease, as well as the extrema, we need the first derivative $f'(x)$.

First derivative of $f(x)$:

Using the quotient rule: $f'(x) = \frac{(x^2 + 1)(2) - (2x)(2x)}{(x^2 + 1)^2}$ $f'(x) = \frac{2x^2 + 2 - 4x^2}{(x^2 + 1)^2} = \frac{-2x^2 + 2}{(x^2 + 1)^2}$ $f'(x) = \frac{2(1 - x^2)}{(x^2 + 1)^2}$

Critical points:

We set $f'(x) = 0$: $\frac{2(1 - x^2)}{(x^2 + 1)^2} = 0$ This occurs when: $1 - x^2 = 0 \quad \Longrightarrow \quad x^2 = 1 \quad \Longrightarrow \quad x = \pm 1$

Thus, the critical points are $x = 1$ and $x = -1$.

Increasing and decreasing intervals:

• Sign of $f'(x)$: $f'(x) = \frac{2(1 - x^2)}{(x^2 + 1)^2}$
  • For $x \in (-\infty, -1) \cup (1, \infty)$, $1 - x^2 < 0 \Rightarrow f'(x) < 0$: the function is decreasing.
  • For $x \in (-1, 1)$, $1 - x^2 > 0 \Rightarrow f'(x) > 0$: the function is increasing.

Extrema:

• At $x = -1$ and $x = 1$, the function switches between increasing and decreasing:
  • Local minimum at $x = -1$.
  • Local maximum at $x = 1$.

Summary:

1. Positivity: $x > 0$
2. Negativity: $x < 0$
3. Zero: $x = 0$
4. Increasing: $(-1, 1)$
5. Decreasing: $(-\infty, -1) \cup (1, \infty)$
6. Local extrema:
   • Local maximum at $x = 1$.
   • Local minimum at $x = -1$.

Would you like to dive deeper into any step or see the graph of the function for more clarity?

Related Questions:

1. How do we confirm the nature of extrema using the second derivative?
2. What happens to the function as $x \to \infty$ or $x \to -\infty$?
3. Can the function have any asymptotes? If so, what kind?
4. How does the behavior of this function compare to $f(x) = \frac{x}{x^2 + 1}$?
5. How do we find concavity and inflection points for this function?

Tip: Always remember that the denominator of a rational function can help you determine the overall behavior, especially for signs and limits.