The exercise on the board contains two main parts related to calculus and function analysis:

1. Critical Points and Intervals of Increase/Decrease

The first question asks to determine the critical points and the intervals where the function $f(x)$ is increasing or decreasing. There are two subproblems:

• (a) $f(x) = \frac{x}{x^2 + 3}$
• (b) $f(x) = x^3(4 - x)$

To solve this:

• Find the first derivative $f'(x)$ of each function.
• Set $f'(x) = 0$ to find the critical points.
• Determine the sign of $f'(x)$ on intervals around the critical points to identify where the function is increasing or decreasing.

2. Maximum/Minimum Points of a Function

The second question requires identifying maximum or minimum points of a function $f$ if they exist, with several sub-functions:

• (a) $f(x) = x \sqrt{x^2 + 4}$
• (b) $f(x) = x \sqrt{x - 4}$
• (c) $f(x) = \frac{x}{\sqrt{x + 1}}$
• (d) $f(x) = \sqrt{x^2 + 4}$
• (e) $f(x) = x (2 - x^2)$

For each subproblem:

• Compute $f'(x)$ and find points where $f'(x) = 0$.
• Analyze the sign changes of $f'(x)$ around these points to identify maximum or minimum values.

Would you like a detailed solution to each subproblem?

Related Questions to Explore Further

1. What is the general process for finding the intervals of increase and decrease for any function?
2. How can we identify and interpret critical points geometrically?
3. What is the difference between local and global maxima/minima in a function?
4. How can the second derivative help in determining the concavity and points of inflection of a function?
5. Why is it important to consider the domain of the function when finding critical points?

Tip

Always verify if the critical points lie within the domain of the function, as functions with restricted domains may have critical points that are not valid.