$$\because f: R \rightarrow R \text { and } f(0)=0, f(1)=1, f(2)=-1 \text {, }$$

$$f(3)=2$$ and $$f(4)=-2$$ then

$$f(x)$$ has atleast 4 real roots.

Then $$f(x)$$ has atleast 3 real roots and $$f^{\prime}(x)$$ has atleast 2 real roots.

Now we know that

$$\begin{aligned} \frac{d}{d x}\left(f^3 \cdot f^{\prime \prime}\right) & =3 f^2 \cdot f^{\prime} \cdot f^{\prime \prime}+f^3 \cdot f^{\prime \prime \prime} \\ & =f^2\left(3 f^{\prime} \cdot f^{\prime}+f \cdot f^{\prime \prime}\right) \end{aligned}$$

Here $$f^3 \cdot f'$$ has atleast 6 roots.

Then its differentiation has atleast 5 distinct roots.