Question

If $f\left(x\right)=\{\begin{array}{cc}{x}^2+2& x<0\\ 3& x=0\\ x+2& x>0\end{array}$, then which of the following statement(s) is/are false ?

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

Answer 1

Answer: (D)

$f^{\prime }\left(x\right)$ is strictly increasing on the right of $x=0$

$f\left(0\right)=3$
$\underset{x\to 0^{-}}{lim}f\left(x\right)=2$
$\underset{x\to 0^{+}}{lim}f\left(x\right)=2$
Therefore there is a local maximum at $x=0$
$f^{\prime }\left(x\right)=2x$ for $x<0$ therefore $f\left(x\right)$ is decreasing for $x<0$
$f^{\prime \prime }\left(x\right)=2$ for $x<0$, hence $f^{\prime }\left(x\right)$ is increasing for $x<0$
$f^{\prime }\left(x\right)=1>0forx>0$ therefore $f\left(x\right)$ is increasing for $x>0$
$f^{\prime }\left(x\right)$ is constant for $x>0$ , hence option D is false