To find the number of distinct real roots of the equation \(3x^4 + 4x^3 - 12x^2 + 4 = 0\), we will analyze the function and its derivative step by step.
### Step 1: Define the function
Let \(f(x) = 3x^4 + 4x^3 - 12x^2 + 4\).
### Step 2: Find the derivative
To determine the nature of the roots, we first find the derivative of \(f(x)\):
\[
f'(x) = \frac{d}{dx}(3x^4 + 4x^3 - 12x^2 + 4) = 12x^3 + 12x^2 - 24x
\]
### Step 3: Factor the derivative
We can factor out the common term from the derivative:
\[
f'(x) = 12x(x^2 + x - 2)
\]
Next, we factor the quadratic:
\[
x^2 + x - 2 = (x + 2)(x - 1)
\]
Thus, we have:
\[
f'(x) = 12x(x + 2)(x - 1)
\]
### Step 4: Find critical points
Setting the derivative equal to zero gives us the critical points:
\[
12x(x + 2)(x - 1) = 0
\]
The critical points are:
\[
x = 0, \quad x = -2, \quad x = 1
\]
### Step 5: Analyze the intervals
We will analyze the sign of \(f'(x)\) in the intervals determined by the critical points: \((-∞, -2)\), \((-2, 0)\), \((0, 1)\), and \((1, ∞)\).
1. **Interval \((-∞, -2)\)**: Choose \(x = -3\):
\[
f'(-3) = 12(-3)(-1)(-4) < 0 \quad \text{(decreasing)}
\]
2. **Interval \((-2, 0)\)**: Choose \(x = -1\):
\[
f'(-1) = 12(-1)(1)(-2) > 0 \quad \text{(increasing)}
\]
3. **Interval \((0, 1)\)**: Choose \(x = 0.5\):
\[
f'(0.5) = 12(0.5)(2.5)(-0.5) < 0 \quad \text{(decreasing)}
\]
4. **Interval \((1, ∞)\)**: Choose \(x = 2\):
\[
f'(2) = 12(2)(4)(1) > 0 \quad \text{(increasing)}
\]
### Step 6: Determine the nature of the function
From the analysis, we see that:
- \(f(x)\) is decreasing on \((-∞, -2)\),
- increasing on \((-2, 0)\),
- decreasing on \((0, 1)\),
- increasing on \((1, ∞)\).
### Step 7: Evaluate the function at critical points
Now we evaluate \(f(x)\) at the critical points to determine the number of roots:
1. **At \(x = -2\)**:
\[
f(-2) = 3(-2)^4 + 4(-2)^3 - 12(-2)^2 + 4 = 48 - 32 - 48 + 4 = -28 < 0
\]
2. **At \(x = 0\)**:
\[
f(0) = 4 > 0
\]
3. **At \(x = 1\)**:
\[
f(1) = 3(1)^4 + 4(1)^3 - 12(1)^2 + 4 = 3 + 4 - 12 + 4 = -1 < 0
\]
### Step 8: Conclusion on the number of distinct real roots
From the evaluations:
- \(f(-2) < 0\) and \(f(0) > 0\) indicates a root in \((-2, 0)\).
- \(f(0) > 0\) and \(f(1) < 0\) indicates a root in \((0, 1)\).
- Since \(f(1) < 0\) and as \(x\) approaches \(+\infty\), \(f(x) \to +\infty\), there is another root in \((1, +\infty)\).
Thus, we conclude that there are **three distinct real roots** of the equation \(3x^4 + 4x^3 - 12x^2 + 4 = 0\).
