Piecewise function differentiability

Let $h:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two bounded functions. Define the function $f:\mathbb{R}\to\mathbb{R}$ by $$f(x)=\begin{cases}xg(x)+x^{2}h(x),&x>0,\\cx+dx^{2},&x\le 0,\end{cases}$$ where $c,d$ are real constants. Consider the following statements. $P$: If $c=g(0)=h(0)$, then $f$ is differentiable at $x=0$. $Q$: If $f$, $g$, and $h$ are thrice differentiable functions with $g(0)=0$ and $g'(0)+h(0)>0$, then $f$ has a local minimum at $x=0$. Then