Let `f(x)=x^3-3(a-2)x^2+3ax+7` and `f(x)` is increasing in `(0,1]` and decreasing is `[1,5)`, then roots of the equation `(f(x)-14)/((x-1)^2)=0` is (A) `1` (B) `3` (C) `7` (D) `-2`

Options

A. 6
B. 5
C. 7
D. -7

Answer

To solve the problem, we need to analyze the function \( f(x) = x^3 - 3(a - 2)x^2 + 3ax + 7 \) and determine the conditions under which it is increasing and decreasing.

### Step 1: Find the derivative of \( f(x) \)
To determine where \( f(x) \) is increasing or decreasing, we first find its derivative:

\[ f'(x) = \frac{d}{dx}(x^3 - 3(a - 2)x^2 + 3ax + 7) \]
\[ = 3x^2 - 6(a - 2)x + 3a \]

### Step 2: Set the derivative to zero to find critical points
Next, we set the derivative equal to zero to find critical points:

\[ 3x^2 - 6(a - 2)x + 3a = 0 \]
Dividing the entire equation by 3 gives:

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

### Step 3: Analyze the intervals of increase and decrease
We know that \( f(x) \) is increasing on \( (0, 1] \) and decreasing on \( [1, 5) \). This means that:

1. The critical point at \( x = 1 \) must be where the function changes from increasing to decreasing.
2. Therefore, \( f'(1) = 0 \).

Substituting \( x = 1 \) into the derivative:

\[ 1^2 - 2(a - 2)(1) + a = 0 \]
\[ 1 - 2(a - 2) + a = 0 \]
\[ 1 - 2a + 4 + a = 0 \]
\[ 5 - a = 0 \implies a = 5 \]

### Step 4: Substitute \( a \) back into \( f(x) \)
Now that we have \( a = 5 \), we substitute it back into \( f(x) \):

\[ f(x) = x^3 - 3(5 - 2)x^2 + 3(5)x + 7 \]
\[ = x^3 - 9x^2 + 15x + 7 \]

### Step 5: Solve the equation \( \frac{f(x) - 14}{(x - 1)^2} = 0 \)
We need to find the roots of the equation:

\[ f(x) - 14 = 0 \]
Substituting \( f(x) \):

\[ x^3 - 9x^2 + 15x + 7 - 14 = 0 \]
\[ x^3 - 9x^2 + 15x - 7 = 0 \]

### Step 6: Factor the polynomial
To find the roots, we can use synthetic division or factorization. By testing \( x = 1 \):

\[ 1^3 - 9(1^2) + 15(1) - 7 = 1 - 9 + 15 - 7 = 0 \]

Thus, \( x = 1 \) is a root. Now we can factor \( (x - 1) \) out:

Using synthetic division:

\[ \begin{array}{r|rrrr} 1 & 1 & -9 & 15 & -7 \\ & & 1 & -8 & 7 \\ \hline & 1 & -8 & 7 & 0 \\ \end{array} \]

This gives us:

\[ x^3 - 9x^2 + 15x - 7 = (x - 1)(x^2 - 8x + 7) \]

### Step 7: Factor the quadratic
Now we need to factor \( x^2 - 8x + 7 \):

\[ x^2 - 8x + 7 = (x - 1)(x - 7) \]

Thus, the complete factorization is:

\[ (x - 1)^2(x - 7) = 0 \]

### Step 8: Find the roots
The roots are:

\[ x - 1 = 0 \implies x = 1 \quad \text{(double root)} \]
\[ x - 7 = 0 \implies x = 7 \]

### Conclusion
The roots of the equation \( \frac{f(x) - 14}{(x - 1)^2} = 0 \) are \( x = 1 \) and \( x = 7 \). The answer to the question is:

**(C) 7**