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  4. If the function f given by f(x) = x3 -3(a - 2)x 2 + 3ax + 7, for some a∈ R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, (f(x)-14/(x-1)2) = 0(x ≠ 1) is:

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Q. If the function $f$ given by $f(x) = x^3 -3(a - 2)x ^2 + 3ax + 7$, for some a$\in R$ is increasing in $(0, 1]$ and decreasing in $[1, 5)$, then a root of the equation, $\frac{f(x)-14}{(x-1)^2} = 0(x \ne 1) is:$

JEE MainJEE Main 2019Application of Derivatives

Solution:

$f'(x) = 3x^2 - 6(a - 2)x + 3a$
$f'(x) \, \ge \, 0 \forall \, x \, \in (0, 1]$
$f'(x) \le \, 0 \forall \, x \, \in \, [1,5)$
$\Rightarrow \, \, \, f'(x) \, -14 \, = (x-1)^2 \, (x-7)$
$\frac{f(x) -14}{(x-1)^2} \, = x-7$