Question

Let $f\left(x\right)=\sin x+e^x+x,$ then which of the following(s) is(are) true about $f\left(x\right)$

• A. increasing for all $x<0$
• B. decreasing for all $x\in R-\left\{0\right\}$
• C. increasing at $x=0$
• D. neither increasing nor decreasing at $x=0$

Answer 1

Answer: (A), (C)

increasing at $x=0$

Given: $f\left(x\right)=\sin x+e^x+x$
$\Rightarrow f^{\prime }\left(x\right)=\cos x+e^x+1>0\forall x\in R$
$(\because 1+\cos x\ge 0,e^x>0,\forall x\in R)$
$\therefore f\left(x\right)$ is increasing for all $x\in R.$