Practicing Success
If f: R → R given by $f(x) = x^3+(a + 2) x^2 + 3ax +5$ is one-one, then a belongs to the interval |
(-∞, 1) (1, ∞) (1, 4) (4, ∞) |
(1, 4) |
The correct answer is Option (3) → (1, 4) Since f: R → R is one-one. Therefore, f(x) is either strictly increasing or strictly decreasing. $⇒ f'(x)>0$ or $f'(x) <0$ for all x $⇒3x^2 + 2x (a + 2) + 3a > 0$ for all $x ∈ R$ or, $3x^2 + 2x (a + 2) + 3a <0$ for all $x ∈ R$ $⇒3x^2 + 2x (a + 2) + 3a > 0$ for all x $⇒4 (a + 2)^2 - 36a <0$ [$∵ ax^2+ bx + c>0$ for all x ⇒ Disc <0] $⇒4 (a^2 + 4a + 4-9a) <0$ $⇒(a^2 −5a+4) <0⇒ (a− 1) (a−4) <0⇒ 1 <a<4$ Hence, f(x) is one-one if $a ∈ (1, 4)$. |