Question

If the function $f\left(x\right)=x^{3}-3ax$ has a local minimum at $x=\lambda \, \left(\lambda \geq 4\right)$ and $a\in \left[10,18\right]$ , then the sum of all the possible integral values of $a$ is

• A. $50$
• B. $112$
• C. $51$
• D. $16$

Answer 1

Answer: (C)

$f^{'} \left(x\right) = 3 x^{2} - 3 a$
$=3\left(x - \sqrt{a}\right)\left(x + \sqrt{a}\right)$

$\therefore f\left(x\right)$ has a local minimum at $x=\sqrt{a}$
$\Rightarrow \sqrt{a}\geq 4$
$\Rightarrow a\geq 16$
$\therefore $ $‘a’$ can be $16,17,18$