Discussion
If the function \(f(x) = x^{2} + ax + 1\) is increasing on [1,2] , then \(a\) is greater than or equal to
Answer: A
\(a\) must be greater than or equal to -2.
\(f'(x) = 2x + a\). For \(f\) to be increasing on [1, 2], \(f'(x) \ge 0\) throughout the interval. Since \(2x + a\) is smallest at \(x = 1\), the condition is
\[ 2(1) + a \ge 0 \quad \Rightarrow \quad a \ge -2 \]
Using the other end, \(x = 2\), would give \(a \ge -4\), which is not enough: with \(a = -3\), for example, \(f'(1) = -1 \lt 0\) and \(f\) decreases at the start of the interval.
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