1. Given function

We need the interval on which $f$ is increasing.

2. Find derivative in each region

For $x>0$

Factorizing:

So for $x>0$,

This quadratic is positive between its roots:

Hence,

But here we only consider $x>0$, so

Thus $f$ is increasing on

in the positive region.

For $x\le 0$

Using product rule:

Since $e^x>0$ for all $x$,

But here $x\le 0$, so

Thus $f$ is increasing on

in the non-positive region.

3. Combine the intervals

From above:

• increasing on $(-1,0]$
• increasing on $(0,\tfrac32)$

So overall, the function is increasing on

(Indeed, at $x=0$, both pieces join continuously since $f(0)=0$, and the increasing nature continues across $0$.)

4. Check options

• A: $\left(-\frac12,2\right)$ is not correct because for $x>\frac32$, $f'(x)<0$.
• B: $(0,2)$ is not correct because again for $x>\frac32$, $f'(x)<0$.
• C: $\left(-1,\frac32\right)$ is correct.
• D: $(-3,-1)$ is not correct because for $x<-1$, $f'(x)<0$.

5. Final answer

The correct option is