Let $$f : \mathbb{R} \to \mathbb{R}$$ be a thrice differentiable function such that $$f(0) = 0, f(1) = 1, f(2) = -1, f(3) = 2$$ and $$f(4) = -2$$. Then, the minimum number of zeros of $$(3f'f'' + ff''')(x)$$ is _____

$$g(x)$$ looks like a part of a higher derivative.

Let $$h(x) = f(x) \cdot f''(x) + (f'(x))^2$$ is the derivative of $$f \cdot f'$$.

The correct observation: $$g(x)$$ is the derivative of $$f(x) \cdot f''(x) + (f'(x))^2$$... almost.

, $$(f \cdot f'')' = f' f'' + f f'''$$.

Let $$k(x) = f(x) f''(x)$$. By Rolle's theorem on $$f(x)$$:

$$f(0)=0, f(1)=1, f(2)=-1, f(3)=2, f(4)=-2$$.

$$f(x)$$ has 4 zeros in $$(0, 4)$$ by Intermediate Value Theorem.

Total zeros of $$f(x) = 5$$ (including $$x=0$$).

Following the chain of derivatives through Rolle's Theorem, the minimum zeros for this specific expression is 5