Question

If $f\left(x\right)=\sin x+a^{2}x+b$ is an increasing function for all values of x, then

• A. $aϵ(-\infty ,-1)$
• B. $aϵR$
• C. $aϵ(-1,1)$
• D. none of these

Answer 1

The correct option is A $aϵ(-\infty ,-1)$

For a function $f\left(x\right)$ to be increasing, $f^{{}^{\prime }}\left(x\right)>0$

$f\left(x\right)=sinx+a^{2}x+b$

$\therefore f^{{}^{\prime }}\left(x\right)=cosx+a^2>0$

$-1+a^2\le cosx+a^2\le 1+a^2$

If $a^2-1>0$, then $cosx+a^2>0$ $\forall x$

$implies-1+a^2>0$

$a\in (-\infty ,-1)\cup (1,\infty )$

So, the answer is option (A)