Consider the following function
$$ f(x) = \left\{ \begin{array}{ll} \left|x\right| & \quad x <2\\ \lfloor x\rfloor & \quad x \ge 2 \end{array} \right. $$
where $\lfloor\rfloor$ denotes the greatest integer function. For $x \in(-1,4)$ I need to analyze which options are true ?
(a) $f$ has a local maxima
(b) $f$ has a local minima
(c) $f$ is continuous
(d) $f$ is not differentiable
So, option (c) is false as the greatest integer is discontinuous at integral points.
option (d) is correct as $|x|$ is non differentiable at $x=0$ also , $\lfloor x\rfloor$ is non differentiable at integral points.
(b) Is correct because $f(0-) > f(0)$ and $f(0) < f(0+)$ so $f$ has a local minima at $x=0$
Option (a) also seems to be correct because
for $x \ge 2$ $f = \lfloor x\rfloor$ and
$f(3-) = 2$ while $f(3) = f(3+) =3$ so $f(3-) < f(3)$ and $f(3) \ge f(3+)$ so $x =3$ is one such point of local maxima.
So, according to me options (a) (b) and (d) are correct.
However, my book says correct choices are (b) and (d)
Can anyone please explain why is option (a) incorrect ? and what is wrong with my solution ?
Thank you.