Let's analyze each question step-by-step.

Question 12

To find the differential of the function $f(x) = 2x - \frac{3}{x}$:

1. Differentiate $f(x)$ with respect to $x$: $f'(x) = \frac{d}{dx}\left(2x\right) - \frac{d}{dx}\left(\frac{3}{x}\right) = 2 + \frac{3}{x^2}$
2. The differential $df$ is given by $f'(x) \, dx$: $df = \left(2 + \frac{3}{x^2}\right) dx$

Check which option matches this answer.

Question 13

To find intervals of increase and decrease for $f(x) = x^3 - 2x$:

1. Find the derivative $f'(x) = 3x^2 - 2$.
2. Set $f'(x) = 0$ to find critical points: $3x^2 - 2 = 0 \Rightarrow x = \pm \sqrt{\frac{2}{3}}$
3. Use test points in each interval around $x = \pm \sqrt{\frac{2}{3}}$ to determine whether $f(x)$ is increasing or decreasing.

Compare with the given options.

Question 14

To find the relative maxima and minima for $f(x) = \frac{2}{3}x^3 + \frac{5}{2}x^2 - 12x + 7$:

1. Compute $f'(x)$ and set it to zero to find critical points.
2. Use the second derivative test to classify each critical point as a maximum or minimum.
3. Substitute critical points back into $f(x)$ to get the function values.

Again, compare with the options.

Would you like a detailed calculation for each question?

Here are five related questions for further exploration:

1. How do you find the differential of more complex functions?
2. What is the importance of critical points in determining intervals of increase and decrease?
3. How does the second derivative test help in classifying maxima and minima?
4. Can relative extrema exist at endpoints of intervals?
5. How can we interpret the differential geometrically?

Tip: Always check each interval thoroughly when analyzing signs around critical points for increasing/decreasing functions.