Question

If $f:R\to R$ given by $f\left(x\right)=x^3+({a+2)x}^2+3ax+5$ is one-one, then $a$ belongs to the interval

• A. $(-\infty ,1)$
• B. $(1,\infty )$
• C. $(1,4)$
• D. $(4,\infty )$

Answer 1

The correct option is C $(1,4)$

If $f\left(x\right)$ is one one then $f^{\prime }\left(x\right)$ should be either positive or negative for all $x$.

$f^{\prime }\left(x\right)=3x^2+2(a+2)x+3a$

$\Rightarrow D=4{(a+2)}^2-4\times 3\times 3a<0$

$\Rightarrow$ $a^2-5a+4<0$

$\Rightarrow$ $(a-1)(a-4)<0$

Hence, $a\in (1,4)$