Let `f(x)=0` be a polynomial equation with real coefficients. Then between any two distinct real roots of `f(x)=0`, there exists at least one real root of the equation `f'(x)=0`. This result is a consequence of the celebrated Rolle's theorem applied to polynomials. Much information can be extracted about the roots of `f(x)=0` from the roots of `f'(x)=0`.
The range of values of k for which the equation `x^(4)+4x^(3)-8x^(2)+k=0` has four real and unequal roots is

Options

A. `0 lt k lt 3`
B. `0 lt k lt 8`
C. `3 lt k lt 8`
D. `3 lt k lt 13`

Answer

To find the range of values of \( k \) for which the polynomial equation \( f(x) = x^4 + 4x^3 - 8x^2 + k = 0 \) has four real and unequal roots, we will follow these steps:

### Step 1: Find the derivative of \( f(x) \)
The first step is to compute the derivative of the polynomial \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^4 + 4x^3 - 8x^2 + k) = 4x^3 + 12x^2 - 16x \]

### Step 2: Set the derivative equal to zero
Next, we set the derivative equal to zero to find the critical points:
\[ 4x^3 + 12x^2 - 16x = 0 \]
Factoring out the common term:
\[ 4x(x^2 + 3x - 4) = 0 \]
This gives us:
\[ x = 0 \quad \text{or} \quad x^2 + 3x - 4 = 0 \]

### Step 3: Solve the quadratic equation
Now, we solve the quadratic equation \( x^2 + 3x - 4 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{-3 \pm \sqrt{9 + 16}}{2} = \frac{-3 \pm 5}{2} \]
This gives us the roots:
\[ x = 1 \quad \text{and} \quad x = -4 \]

### Step 4: Identify the critical points
The critical points from \( f'(x) = 0 \) are:
\[ x = 0, \quad x = 1, \quad x = -4 \]

### Step 5: Analyze the behavior of \( f(x) \)
To ensure \( f(x) \) has four real and unequal roots, \( f(x) \) must change signs at the critical points. We evaluate \( f(x) \) at these points:
1. \( f(0) = k \)
2. \( f(1) = 1^4 + 4(1)^3 - 8(1)^2 + k = 1 + 4 - 8 + k = -3 + k \)
3. \( f(-4) = (-4)^4 + 4(-4)^3 - 8(-4)^2 + k = 256 - 256 - 128 + k = -128 + k \)

### Step 6: Set up inequalities for four real roots
For \( f(x) \) to have four real and unequal roots, the following conditions must hold:
1. \( f(0) = k > 0 \)
2. \( f(1) = -3 + k < 0 \) implies \( k < 3 \)
3. \( f(-4) = -128 + k < 0 \) implies \( k < 128 \)

### Step 7: Combine the inequalities
From the above conditions, we have:
\[ 0 < k < 3 \]

### Conclusion
Thus, the range of values of \( k \) for which the equation \( x^4 + 4x^3 - 8x^2 + k = 0 \) has four real and unequal roots is:
\[ \boxed{(0, 3)} \]