If the function `f(x) = x^(2) - ax + 5` is strictly increasing on (1,2) , then a lies in the interval

Options

A. `(2 , infty)`
B. `(- infty, 2)`
C. `(4, infty)`
D. `(- infty, 4)`

Answer

To determine the interval in which the value of \( a \) lies such that the function \( f(x) = x^2 - ax + 5 \) is strictly increasing on the interval \( (1, 2) \), we will follow these steps:

### Step 1: Find the derivative of the function
The first step is to find the derivative of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(x^2 - ax + 5) = 2x - a \]

### Step 2: Set the derivative greater than zero
Since the function is strictly increasing on the interval \( (1, 2) \), we need to ensure that the derivative \( f'(x) \) is greater than zero for all \( x \) in this interval.

\[ f'(x) > 0 \implies 2x - a > 0 \]

### Step 3: Solve the inequality for the interval \( (1, 2) \)
We will evaluate the inequality \( 2x - a > 0 \) at the endpoints of the interval \( (1, 2) \).

1. For \( x = 1 \):
\[ 2(1) - a > 0 \implies 2 - a > 0 \implies a < 2 \]

2. For \( x = 2 \):
\[ 2(2) - a > 0 \implies 4 - a > 0 \implies a < 4 \]

### Step 4: Combine the results
From the inequalities derived from the endpoints, we have:
- From \( x = 1 \): \( a < 2 \)
- From \( x = 2 \): \( a < 4 \)

Since we need \( a \) to satisfy both conditions, the more restrictive condition is \( a < 2 \).

### Conclusion
Thus, the value of \( a \) must lie in the interval:

\[ (-\infty, 2) \]