Cho hàm số y = x^3 - 3/4x^2 - 3/2x

To analyze the function \( y = x^3 - \frac{3}{4}x^2 - \frac{3}{2}x \) in the problem, we can follow these steps for the parts (a) and (b):

### a) Examine the derivative and sketch the graph of the function

1. **Find the derivative \( y' \)**:
\[
y' = \frac{d}{dx}(x^3 - \frac{3}{4}x^2 - \frac{3}{2}x) = 3x^2 - \frac{3}{2}x - \frac{3}{2}
\]

2. **Set \( y' = 0 \) to find critical points**:
\[
3x^2 - \frac{3}{2}x - \frac{3}{2} = 0
\]
Multiply through by 2 to clear fractions:
\[
6x^2 - 3x - 3 = 0
\]

Use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 6 \cdot (-3)}}{2 \cdot 6}
\]
\[
= \frac{3 \pm \sqrt{9 + 72}}{12} = \frac{3 \pm \sqrt{81}}{12} = \frac{3 \pm 9}{12}
\]
So the critical points are:
\[
x_1 = 1, \quad x_2 = -\frac{1}{2}
\]

3. **Determine the nature of these critical points** by evaluating the second derivative \( y'' \):
\[
y'' = \frac{d}{dx}(3x^2 - \frac{3}{2}x - \frac{3}{2}) = 6x - \frac{3}{2}
\]

Evaluate at \( x = 1 \) and \( x = -\frac{1}{2} \):
- For \( x = 1 \):
\[
y''(1) = 6(1) - \frac{3}{2} = 6 - 1.5 = 4.5 \quad (\text{local minimum})
\]
- For \( x = -\frac{1}{2} \):
\[
y''\left(-\frac{1}{2}\right) = 6 \left(-\frac{1}{2}\right) - \frac{3}{2} = -3 - 1.5 = -4.5 \quad (\text{local maximum})
\]

4. **Find the function values at these points**:
- For \( x = 1 \):
\[
y(1) = 1^3 - \frac{3}{4}(1^2) - \frac{3}{2}(1) = 1 - \frac{3}{4} - \frac{3}{2} = 1 - 0.75 - 1.5 = -1.25
\]
- For \( x = -\frac{1}{2} \):
\[
y\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^3 - \frac{3}{4}\left(-\frac{1}{2}\right)^2 - \frac{3}{2}\left(-\frac{1}{2}\right)
\]
\[
= -\frac{1}{8} - \frac{3}{16} + \frac{3}{4} = -\frac{1}{8} - \frac{3}{16} + \frac{12}{16} = -\frac{2}{16} + \frac{12}{16} = \frac{10}{16} = \frac{5}{8}
\]

5. **Summarize the behavior**:
- Local maximum at \( \left(-\frac{1}{2}, \frac{5}{8}\right) \)
- Local minimum at \( (1, -1.25) \)

### b) Analyze the equation \( |x|^3 - 3x^2 - 6|x| - 4a = 0 \)

1. **Investigate the equation**:
- Treat \( |x| \) as a variable, say \( z = |x| \), then rewrite:
\[
z^3 - 3z^2 - 6z - 4a = 0
\]

2. **Determine the number of solutions**:
- Use the derivative \( g'(z) = 3z^2 - 6z - 6 \) to find turning points.
- Examine the discriminant of \( 3z^2 - 6z - 6 = 0 \) to see where it has real roots.

3. **Find critical points of \( g(z) \)** where the nature of the polynomial changes.
- Solve \( 3z^2 - 6z - 6 = 0 \) using the quadratic formula and then use test values around these points to determine the behavior and number of intersections.

Given these analyses, you would conclude the nature of the function and the behavior of the equation accordingly.