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)-14x^(2)+24x-k=90` has four unequal real roots is

Options

A. `8 lt k lt 11`
B. `4 lt k lt 8`
C. `8 lt k lt 15`
D. `4 lt k lt 13`

Answer

To solve the problem, we need to find the range of values of \( k \) for which the polynomial equation

\[ x^4 - 14x^2 + 24x - k = 90 \]

has four distinct real roots. We can rewrite this as:

\[ x^4 - 14x^2 + 24x - k - 90 = 0 \]

Let’s denote the polynomial as:

\[ g(x) = x^4 - 14x^2 + 24x - (k + 90) \]

### Step 1: Find the derivative of \( g(x) \)

To apply Rolle's theorem, we first need to find the critical points of \( g(x) \) by calculating its derivative:

\[ g'(x) = 4x^3 - 28x + 24 \]

### Step 2: Set the derivative to zero

Next, we set the derivative equal to zero to find the critical points:

\[ 4x^3 - 28x + 24 = 0 \]

Dividing through by 4 gives:

\[ x^3 - 7x + 6 = 0 \]

### Step 3: Factor the cubic equation

We can factor this cubic equation. By testing possible rational roots, we find that \( x = 1 \) is a root:

\[ 1^3 - 7(1) + 6 = 0 \]

Using synthetic division or polynomial long division, we can factor \( x^3 - 7x + 6 \) as:

\[ (x - 1)(x^2 + x - 6) = 0 \]

Next, we factor the quadratic:

\[ x^2 + x - 6 = (x - 2)(x + 3) \]

Thus, the complete factorization is:

\[ (x - 1)(x - 2)(x + 3) = 0 \]

### Step 4: Identify the critical points

The critical points are:

\[ x = 1, \quad x = 2, \quad x = -3 \]

### Step 5: Analyze the behavior of \( g(x) \)

To have four distinct real roots for \( g(x) = 0 \), \( g(x) \) must have local maxima and minima. We need to evaluate \( g(x) \) at the critical points:

1. **At \( x = 1 \)**:
\[ g(1) = 1^4 - 14(1^2) + 24(1) - (k + 90) = 1 - 14 + 24 - (k + 90) = 11 - (k + 90) = -k - 79 \]

2. **At \( x = 2 \)**:
\[ g(2) = 2^4 - 14(2^2) + 24(2) - (k + 90) = 16 - 56 + 48 - (k + 90) = 8 - (k + 90) = -k - 82 \]

3. **At \( x = -3 \)**:
\[ g(-3) = (-3)^4 - 14(-3)^2 + 24(-3) - (k + 90) = 81 - 126 - 72 - (k + 90) = -117 - (k + 90) = -k - 207 \]

### Step 6: Determine conditions for four distinct roots

For \( g(x) \) to have four distinct real roots, the values of \( g(1) \), \( g(2) \), and \( g(-3) \) must satisfy the following conditions:

1. \( g(1) > 0 \) (local maximum)
2. \( g(2) < 0 \) (local minimum)
3. \( g(-3) < 0 \) (local minimum)

From \( g(1) > 0 \):
\[ -k - 79 > 0 \implies k < -79 \]

From \( g(2) < 0 \):
\[ -k - 82 < 0 \implies k > -82 \]

From \( g(-3) < 0 \):
\[ -k - 207 < 0 \implies k > -207 \]

### Step 7: Combine the inequalities

The range of \( k \) for which the polynomial has four distinct real roots is:

\[ -82 < k < -79 \]

### Final Answer

The range of values of \( k \) for which the equation \( x^4 - 14x^2 + 24x - k = 90 \) has four unequal real roots is:

\[ \boxed{(-82, -79)} \]