The image shows a calculus worksheet with a focus on concavity and points of inflection. The main questions in the problem set are based on analyzing the second derivative of given functions. Here's a breakdown of the questions:

1. For the function $f(x) = \frac{x}{x^2 + 3}$:

   • (a) What is the domain of $f(x)$?
   • (b) When does $f''(x) = 0$ or not exist?
   • (c) When is $f(x)$ concave up? Concave down?
   • (d) Find all points of inflection on $f(x)$.

2. For the function $f(x) = x^{4/3} + 16x^{1/3}$:

   • Determine concavity and points of inflection.

Let me guide you through the steps to solve these problems.

1. Function: $f(x) = \frac{x}{x^2 + 3}$

• (a) Domain of $f(x)$: The function $f(x) = \frac{x}{x^2 + 3}$ has no restrictions on the domain because $x^2 + 3 \neq 0$ for all real $x$. Therefore, the domain is all real numbers: $(-\infty, \infty)$.

• (b) Second derivative: To find when $f''(x) = 0$ or does not exist, we need to first compute the first and second derivatives of $f(x)$.

  First derivative $f'(x)$ using the quotient rule: $f'(x) = \frac{(x^2 + 3)(1) - x(2x)}{(x^2 + 3)^2} = \frac{3 - x^2}{(x^2 + 3)^2}$

  Second derivative $f''(x)$: To find the second derivative, we differentiate $f'(x)$ again using the quotient rule.

• (c) Concavity: Determine the sign of $f''(x)$ to find intervals where the function is concave up or concave down.

• (d) Points of inflection: Find the values of $x$ where $f''(x) = 0$ or $f''(x)$ changes sign to determine the points of inflection.

Would you like me to compute the derivatives step-by-step and fully solve these? Let me know if you want further clarification!

Relative Questions:

1. How is concavity determined from the second derivative of a function?
2. What is the significance of the points where $f''(x) = 0$ or $f''(x)$ does not exist?
3. Can a function have concavity changes without points of inflection? Why or why not?
4. How does the quotient rule help in finding derivatives of rational functions?
5. What is the relationship between the first derivative and critical points for concavity?

Tip:

Remember, concavity describes how the slope of a function is changing: concave up means the slope is increasing, while concave down means the slope is decreasing.