The values of ‘a’ for which the function (a + 2) x^3 - 3a x^2 + 9… - Vidyadip

MCQ

The values of $ ‘a’$ for which the function $(a + 2){x^3} - 3a{x^2} + 9ax - 1$ decreases monotonically throughout for all real $ x,$ are

A
$a < - 2$
B
$a > - 2$
C
$ - 3 < a < 0$
✓
$ - \infty < a \le - 3$

✓

Answer

Correct option: D.

$ - \infty < a \le - 3$

d
(d) If $f(x) = (a + 2){x^3} - 3a{x^2} + 9ax - 1$ decreases monotonically for all $x \in R,$

then $f'(x) \le 0$ for all $x \in R$

==> $3(a + 2){x^2} - 6ax + 9a \le 0$ for all $x \in R$

==> $(a + 2){x^2} - 2ax + 3a \le 0$ for all $x \in R$

==> $a + 2 < 0$ and Discriminant$ \le 0$

==> $a < - 2$,$ - 8{a^2} - 24a \le 0$ ==> $a < - 2$ and $a(a + 3) \ge 0$

==> $a < - 2$, $a \le - 3$ or $a \ge 0$==> $a \le - 3$==>$ - \infty < a \le - 3$ .

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