If the function f given by


If the function $f$ given by $f(x)=x^{3}-3(a-2) x^{2}+$ $3 a x+7$, for some $a \in R$ is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation,

  1. 6

  2. 5

  3. 7

  4. -7

Correct Option: , 3


$f^{\prime}(x)=3 x^{2}-6(a-2) x+3 a$

$\mathrm{f}^{\prime}(\mathrm{x}) \geq 0 \forall \mathrm{x} \in(0,1]$

$\mathrm{f}^{\prime}(\mathrm{x}) \leq 0 \forall \mathrm{x} \in[1,5)$

$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=0$ at $\mathrm{x}=1 \Rightarrow \mathrm{a}=5$



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