The function `f (x)=2 log (x - 2) - x^2 + 4x +1` increases on the interval

Options

A. `(1,2)`
B. `(2,3)`
C. `(5//2,3)`
D. `(2,4)`

Answer

To determine the interval where the function \( f(x) = 2 \log(x - 2) - x^2 + 4x + 1 \) is increasing, we need to follow these steps:

### Step 1: Find the derivative of the function
To find where the function is increasing, we first need to compute the derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx} \left( 2 \log(x - 2) - x^2 + 4x + 1 \right) \]

Using the chain rule and the properties of logarithmic differentiation, we have:

\[ f'(x) = 2 \cdot \frac{1}{x - 2} \cdot (1) - 2x + 4 \]

So,

\[ f'(x) = \frac{2}{x - 2} - 2x + 4 \]

### Step 2: Set the derivative greater than or equal to zero
To find the intervals where the function is increasing, we set the derivative greater than or equal to zero:

\[ \frac{2}{x - 2} - 2x + 4 \geq 0 \]

### Step 3: Simplify the inequality
Rearranging gives:

\[ \frac{2}{x - 2} \geq 2x - 4 \]

Multiplying through by \( x - 2 \) (noting that \( x - 2 > 0 \) for \( x > 2 \)):

\[ 2 \geq (2x - 4)(x - 2) \]

Expanding the right side:

\[ 2 \geq 2x^2 - 8x + 8 \]

Rearranging gives:

\[ 0 \geq 2x^2 - 8x + 6 \]

Dividing through by 2:

\[ 0 \geq x^2 - 4x + 3 \]

### Step 4: Factor the quadratic
Factoring the quadratic:

\[ 0 \geq (x - 1)(x - 3) \]

### Step 5: Determine the intervals from the inequality
The critical points are \( x = 1 \) and \( x = 3 \). We analyze the sign of the product \( (x - 1)(x - 3) \):

- For \( x < 1 \): both factors are negative, product is positive.
- For \( 1 < x < 3 \): first factor is positive, second is negative, product is negative.
- For \( x > 3 \): both factors are positive, product is positive.

Thus, the function is increasing in the interval \( [1, 3] \).

### Step 6: Consider the domain of the original function
Since \( f(x) \) involves \( \log(x - 2) \), we need \( x - 2 > 0 \) or \( x > 2 \). Therefore, we must also consider this condition.

### Step 7: Find the intersection of the intervals
The function is increasing in the interval \( [1, 3] \) and is defined for \( x > 2 \). The intersection of these intervals is:

\[ (2, 3] \]

### Final Answer
The function \( f(x) \) increases on the interval \( (2, 3] \).

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