The greatest possible value of 'a' such that the function f(x) = x^2+…

The correct answer is Option (2) → -4

Given function:

$f(x) = x^{2} + ax + 1$

For $f(x)$ to be always decreasing on $[1,2]$:

$f'(x) \le 0$ for all $x \in [1,2]$

$f'(x) = 2x + a$

The derivative is largest at $x = 2$ .

So the condition must hold at $x = 2$:

$2(2) + a \le 0$

$4 + a \le 0$

$a \le -4$

Thus the greatest possible value of $a$ is:

$-4$