Which of the  statements is true for $f(x)=(x-1)^3(x-2)^2$?

A, f(x) decreases on $(-∞, \frac{8}{5})$

B. f(x) increases on $\left[\frac{8}{5}, 2\right]$

C. f(x) has point of inflection at x=1

D. f(x) decreases on [2,∞)

E. f(x) has its local minimum value at $x=\frac{8}{5}$

Choose the correct answer from the options given below :

C only

The correct answer is Option (2) → C only

$f'(x)=3(x-1)^2(x-2)^2+2(x-1)^3(x-2)$

$=(x-1)^2(x-2)(3x-6+2x-2)$

$=(x-1)^2(x-2)(5x-8)=0$

$⇒x=1,2,\frac{8}{5}$

using wavy curve method

at $x=1$ point of inflection exist

$x=\frac{8}{5}$ (local maxima)

$x=2$ (local minima)

f increases in (-∞,\frac{8}{5}]∪[2,∞)$

f decreases in $[\frac{8}{5},2]$

Only C is correct