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(C) We are given the expression $g(x) = (3 f^{\prime} f^{\prime \prime} + f f^{\prime \prime \prime})(x)$.Note that $\frac{d}{dx} (f(x) f^{\prime}(x))

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$5$

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(C) We are given the expression $g(x) = (3 f^{\prime} f^{\prime \prime} + f f^{\prime \prime \prime})(x)$.Note that $\frac{d}{dx} (f(x) f^{\prime}(x)) = (f^{\prime}(x))^2 + f(x) f^{\prime \prime}(x)$.Also,$\frac{d^2}{dx^2} (f(x) f^{\prime}(x)) = \frac{d}{dx} ((f^{\prime}(x))^2 + f(x) f^{\prime \prime}(x)) = 2 f^{\prime}(x) f^{\prime \prime}(x) + f^{\prime}(x) f^{\prime \prime}(x) + f(x) f^{\prime \prime \prime}(x) = 3 f^{\prime}(x) f^{\prime \prime}(x) + f(x) f^{\prime \prime \prime}(x)$.Thus,the given expression is the second derivative of $h(x) = f(x) f^{\prime}(x)$,i.e.,$g(x) = h^{\prime \prime}(x)$.Given $f(0)=0, f(1)=1, f(2)=-1, f(3)=2, f(4)=-2$,the function $f(x)$ has at least $4$ roots in $(0, 4)$ by the Intermediate Value Theorem (at $x=0$,and between $(1,2), (2,3), (3,4)$).Let $h(x) = f(x) f^{\prime}(x)$. The roots of $h(x)$ are the roots of $f(x)$ and the roots of $f^{\prime}(x)$.$f(x)$ has roots at $x_1=0$,and $x_2 \in (1,2)$,$x_3 \in (2,3)$,$x_4 \in (3,4)$. So $f(x)$ has at least $4$ roots.By Rolle's Theorem,$f^{\prime}(x)$ has at least $3$ roots in $(0, 4)$ (between the roots of $f(x)$).Thus,$h(x) = f(x) f^{\prime}(x)$ has at least $4+3=7$ roots in $[0, 4]$.By Rolle's Theorem,if $h(x)$ has $7$ roots,then $h^{\prime}(x)$ has at least $6$ roots,and $h^{\prime \prime}(x)$ has at least $5$ roots.

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

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