If the function f(x)=x3-3ax has a local minimum at x=λ (λ ≥ 4) and a∈ [10,18] , then the sum of all the possible integral values of a is

Question

If the function f(x)=x3-3ax has a local minimum at x=λ (λ ≥ 4) and a∈ [10,18] , then the sum of all the possible integral values of a is (A) 50 (B) 112 (C) 51 (D) 16

Tardigrade Admin · Accepted Answer

f' (x) = 3 x2 - 3 a =3(x - √a)(x + √a) <img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-uszjtrpl0519.png" /> ∴ f(x) has a local minimum at x=√a ⇒ √a≥ 4 ⇒ a≥ 16 ∴ ‘a’ can be 16,17,18

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

$50$

$112$

$51$

$16$

$f^{^{'}} (x) = 3 x^{2} - 3 a$
$= 3 (x - a) (x + a)$

$∴ f (x)$ has a local minimum at $x = a$
$\Rightarrow a \geq 4$
$\Rightarrow a \geq 16$
$∴$ $‘ a ’$ can be $16, 17, 18$

Q. If the function f(x)=x3−3ax has a local minimum at x=λ(λ≥4) and a∈[10,18] , then the sum of all the possible integral values of a is

50

112

51

16