Question

The value of a for which the function

$(a + 2)x^{3} - 3ax^{2} + 9ax - 1$ decrease monotonically throughout for all real x, are

• A. $a < - 2$
• B. $a > - 2$
• C. $- 3 < a < 0$
• D. $- \infty < a \leq - 3$

Answer 1

Answer: (D)

$- \infty < a \leq - 3$

Answer 2

If $f(x) = (a + 2)x^{3} - 3ax^{2} + 9ax - 1$ decreases monotonically for all $x \in R,$ then $f^{'}(x) \leq 0$ for all $x \in R$

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

⇒ $(a + 2)x^{2} - 2ax + 3a \leq 0$ for all $x \in R$

⇒ $a + 2 < 0$ and discriminant $\leq 0$

⇒ $a < - 2$ and $- 8a^{2} - 24a \leq 0$

⇒ $a < - 2$ and $a(a + 3) \geq 0$⇒ $a < - 2$ and $a \leq - 3$ or $a \geq 0$

⇒ $a \leq - 3$ ⇒ $- \infty < a \leq - 3$