First slide

Functions (XII)

Question

Set of values of $a$ for which the function $f : R \to R$ is given by $f (x) = x^{3} + (a + 2) x^{2} + 3 a x + 10$ is one-one, is given by

Moderate

A

$(- \infty, 1] \cup [4, \infty)$
B

$(1, 4)$
C

$[1, \infty)$
D

$(- \infty, 4)$

Solution

$f : R \to R; f (x) = x^{3} + (a + 2) x^{2} + 3 a x + 10$ is one-one.
Equational process in this is difficult, we use the fact that $f (x)$ is one-one in $R$ , then it must be either increasing in $R$ or decreasing in $R$

$∴ f' (x) = 3 x^{2} + 2 (a + 2) x + 3 a > 0 \forall x \in R$

$< 0 \forall x \in R$

As coefficient of $x^{2} = 3 > 0$ , $f' (x)$ cannot be negative for all $x \in R$

$3 x^{2} + 2 (a + 2) x + 3 a > 0 \forall x \in R$

$\Rightarrow 4 {(a + 2)}^{2} - 36 a < 0$

$\Rightarrow a^{2} - 5 a + 4 < 0$ i.e., $(a - 1) (a - 4) < 0$

$\Rightarrow 1 < a < 4$ i.e., $a \in (1, 4)$ .

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