Question

If $f\left(x\right)=\{\begin{array}{ccc}{x}^2+2& ,& x<0\\ 2& ,& x=0,\\ x+2& ,& x>0\end{array}$ then which of the following is FALSE.

• A. $f\left(x\right)$ has a maximum at $x=0$
• B. $f\left(x\right)$ is strictly decreasing on the left of 0
• C. $f^{\prime }\left(x\right)$ is strictly increasing on the left of 0
• D. $f\left(x\right)$ is strictly increasing on the right of 0

Answer 1

The correct option is A $f\left(x\right)$ has a maximum at $x=0$

$f\left(x\right)=\{\begin{array}{cc}{x}^2+2,& x<0\\ 2& x=0\\ x+2& x>0\end{array}$
Here $f\left(0\right)=2$
${lim}_{x\to 0_{-}}f\left(x\right)=2$
${lim}_{x\to 0_{+}}f\left(x\right)=2$
$f^{\prime }\left(x\right)=2x,x<0$
$\therefore f^{\prime }\left(x\right)<0$ for $x<0$
$\therefore f\left(x\right)$ is decreasing on the left of $0$.
Hence at $x=0$, there is no maximum.
$f"\left(x\right)=2,x<0$
$\therefore f"\left(x\right)>0,x<0$
$\therefore f^{\prime }\left(x\right)$ is increasing on the left of $0$.
$f^{\prime }\left(x\right)=1,x>0$
$\therefore f\left(x\right)$ is strictly increasing on the right of $0$.