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 :
Answer & explanation
Correct answer: option 2
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