The set of values of a 'for which the function f (x) `= x^2 + ax +1` is an increasing function on [1,2] is

Options

A. `(-oo ,-4)`
B. ` [-4,oo)`
C. `[4,oo)`
D. ` (-oo ,4)`

Answer

To determine the set of values of \( a \) for which the function \( f(x) = x^2 + ax + 1 \) is an increasing function on the interval \([1, 2]\), we need to follow these steps:

### Step 1: Understand the condition for increasing function
A function \( f(x) \) is increasing on an interval if its derivative \( f'(x) \) is greater than zero for all \( x \) in that interval.

### Step 2: Differentiate the function
We differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^2 + ax + 1) = 2x + a \]

### Step 3: Set up the inequality
For \( f(x) \) to be increasing on \([1, 2]\), we need:
\[ f'(x) > 0 \quad \text{for all } x \in [1, 2] \]
This leads to the inequality:
\[ 2x + a > 0 \]

### Step 4: Analyze the inequality at the endpoints
We need to check the inequality at the endpoints of the interval:
1. At \( x = 1 \):
\[ 2(1) + a > 0 \implies 2 + a > 0 \implies a > -2 \]
2. At \( x = 2 \):
\[ 2(2) + a > 0 \implies 4 + a > 0 \implies a > -4 \]

### Step 5: Combine the inequalities
The more restrictive condition is \( a > -2 \). Thus, for the function to be increasing on the interval \([1, 2]\), we need:
\[ a > -2 \]

### Conclusion
The set of values of \( a \) for which the function \( f(x) = x^2 + ax + 1 \) is increasing on the interval \([1, 2]\) is:
\[ \boxed{(-2, \infty)} \]