Applications Of Derivatives Question 286
Question: The value of the function $ (x-1){{(x-2)}^{2}} $ at its maxima is
Options:
A) 1
B) 2
C) 0
D) $ \frac{4}{27} $
Show Answer
Answer:
Correct Answer: D
Solution:
Given $ f(x)=(x-1){{(x-2)}^{2}} $
$ f(x)=(x-1)(x^{2}+4-4x) $ ; $ f(x)=(x^{3}-5x^{2}+8x-4) $ Now $ f’(x)=3x^{2}-10x+8 $ , $ f’(x)=0 $
therefore $ 3x^{2}-10x+8=0 $
therefore $ (3x-4)(x-2)=0 $
therefore $ x=\frac{4}{3} $ , 2
Now $ {f}’’(x)=6x-10 $
$ {f}’’(4/3)=6\times 4/3-10<0 $
$ {f}’’(2)=12-10>0 $
Hence at $ x=\frac{4}{3} $ the function will occupy maximum value.
$ \therefore $ Maximum value = $ f(4/3)=4/27 $ .
BETA
