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Step-by-Step Solution
Step 1: Express the function and its derivative
The given function is
f(x) = x^2 + a x + 1 .
To analyze its increasing or decreasing behavior, compute its derivative:
f'(x) = 2x + a .
Step 2: Condition for "f(x) is increasing" on [1, 2]
A function is increasing on an interval if its derivative is nonnegative on that interval. Thus, we need
f'(x) \ge 0 for all x \in [1, 2] .
Substituting f'(x) = 2x + a , we get
2x + a \ge 0 \quad \text{for all } x \in [1, 2].
The most restrictive point in [1, 2] is x = 2 , because 2x is largest there. Hence,
2 \cdot 2 + a \ge 0 \implies 4 + a \ge 0 \implies a \ge -4.
Therefore, the least value of a satisfying this condition is
R = -4.
Step 3: Condition for "f(x) is decreasing" on [1, 2]
A function is decreasing on an interval if its derivative is nonpositive on that interval. Thus, we require
f'(x) \le 0 \quad \text{for all } x \in [1, 2].
Using f'(x) = 2x + a , this becomes
2x + a \le 0 \quad \text{for all } x \in [1, 2].
The most restrictive point in [1, 2] for this inequality is x = 1 , because 2x is smaller there compared to x = 2 . Thus,
2 \cdot 1 + a \le 0 \implies 2 + a \le 0 \implies a \le -2.
Therefore, the greatest value of a satisfying this condition is
S = -2.
Step 4: Compute the required absolute difference
We have found
R = -4 \quad \text{and} \quad S = -2.
Hence,
|R - S| = |-4 - (-2)| = |-4 + 2| = |-2| = 2.
Answer:
|R - S| = 2.
