"Let $f: \mathbb{R} \rightarrow \mathbb{R}$ 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 $\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)$ is __________."

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

$f(3)=2$ 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.

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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ 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 $\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)$ is __________.

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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ 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 $\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)$ is __________.

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