To analyze the function $f(x) = (x^2 + 1)^{\frac{2}{3}}$ and determine its significant features, let's break down the steps:

1. Domain

The function $f(x) = (x^2 + 1)^{\frac{2}{3}}$ is defined for all real values of $x$ because $x^2 + 1$ is always positive (it has no real roots and never becomes negative). Therefore, the domain of $f(x)$ is all real numbers, $x \in (-\infty, \infty)$.

2. Intercepts

• Y-intercept: To find the y-intercept, set $x = 0$. $f(0) = (0^2 + 1)^{\frac{2}{3}} = 1$ So, the y-intercept is at $(0, 1)$.

• X-intercept: To find the x-intercept, set $f(x) = 0$. However, $x^2 + 1 \neq 0$ for any real $x$, so there are no x-intercepts.

3. Symmetry

Since $f(x)$ depends on $x^2$, the function is even: $f(x) = f(-x)$. Thus, the graph is symmetric with respect to the y-axis.

4. Behavior at Infinity

To examine the behavior as $x \to \pm \infty$: $f(x) = (x^2 + 1)^{\frac{2}{3}}$ As $x \to \pm \infty$, $x^2$ dominates the $+1$, so $f(x) \approx (x^2)^{\frac{2}{3}} = |x|^{\frac{4}{3}}$, which grows large as $x$ increases or decreases without bound. Therefore, the function grows unbounded as $x \to \pm \infty$.

5. First Derivative and Critical Points

To find critical points, let's compute the derivative: $f'(x) = \frac{2}{3} (x^2 + 1)^{-\frac{1}{3}} \cdot 2x = \frac{4x}{3 (x^2 + 1)^{\frac{1}{3}}}$ Setting $f'(x) = 0$ gives: $\frac{4x}{3 (x^2 + 1)^{\frac{1}{3}}} = 0 \Rightarrow x = 0$ So, $x = 0$ is a critical point. Since the derivative changes sign around $x = 0$, this is likely an extremum point.

6. Second Derivative for Concavity

To determine concavity, we calculate the second derivative. For simplicity, we can focus on analyzing whether $f(x)$ is concave up or down without finding an exact form for $f''(x)$. The nature of the function suggests that $f(x)$ is concave up for all $x$, given its smooth curve without inflection points.

7. Sketch of the Graph

Based on the above findings:

• The function has a minimum at $(0, 1)$.
• It is symmetric about the y-axis.
• The graph increases as $x \to \pm \infty$ and is concave up throughout its domain.

Would you like further details on any step or have specific questions? Here are related questions to explore further:

1. How would the function change if the exponent were different (e.g., $(x^2 + 1)^{\frac{3}{4}}$)?
2. What happens to the graph if we add a constant to $f(x)$, such as $f(x) + c$?
3. How does the symmetry of even functions affect their critical points and intercepts?
4. How would the derivative test change if there were multiple critical points?
5. Can we use the chain rule to find derivatives of similar functions efficiently?

Tip: When analyzing a function, checking symmetry early can simplify the analysis by reducing the domain you need to consider.