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Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ 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....................
$8$
$4$
$5$
$9$
Solution
$ \left(3 f^{\prime} f^{\prime \prime}+f f f^{\prime \prime \prime}\right)(x)=\left(\left(f f^{\prime \prime}+\left(f^{\prime}\right)^2\right)(x)\right)^{\prime} $
$ \left(\mathrm{ff}^{\prime \prime}+\left(\mathrm{f}^{\prime}\right)^2\right)(\mathrm{x})=\left(\left(\mathrm{ff}^{\prime}\right)(\mathrm{x})\right)^{\prime} $
$ \therefore\left(3 \mathrm{f}^{\prime} \mathrm{f}^{\prime \prime}+\mathrm{f}^{\prime \prime \prime}\right)(\mathrm{x})=\left(\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}^{\prime}(\mathrm{x})\right)^{\prime \prime} $
$Image$
$ \min \text {. roots of } \mathrm{f}(\mathrm{x}) \rightarrow 4 $
$ \therefore \min \text {. roots of } \mathrm{f}^{\prime}(\mathrm{x}) \rightarrow 3 $
$ \therefore \min \text {. roots of }\left(\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}^{\prime}(\mathrm{x})\right) \rightarrow 7 $
$ \therefore \min \text {. roots of }\left(\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}^{\prime}(\mathrm{x})\right)^{\prime \prime} \rightarrow 5$
