QJEEM19Mac716: JEE Main 2019 Mathematics - Class 12 JEEM19Mac716 Solution: If the function f given by f(x) = x 3 – 3(a – 2)x 2 + 3ax + 7,…

Question

Verified Solution • 2025–26 Session moderate

If the function f given by f(x) = x³ – 3(a – 2)x² + 3ax + 7, for some a

\in

R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation,

{{f\left( x \right) - 14} \over {{{\left( {x - 1} \right)}^2}}} = 0\left( {x \ne 1} \right)

is :

Accepted Answer

f '(x) = 3x²

-

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) = 0 at x = 1

\Rightarrow

a = 5

f(x)

-

14 = (x

-

1)² (x

-

7)

{{f(x) - 14} \over {{{\left( {x - 1} \right)}^2}}} = x - 7

Step-by-Step Verified Solution