Applications Of Derivatives Question 243
Question: If $ f(x)=x^{3}-10x^{2}+200x-10 $ , then
[Kurukshetra CEE 1998]
Options:
A) $ f(x) $ is decreasing in $ ]-\infty ,10] $ and increasing in $ [10,,\infty [ $
B) $ f(x) $ is increasing in $ ]-\infty ,10] $ and decreasing in $ [10,,\infty [ $
C) $ f(x) $ is increasing throughout real line
D) $ f(x) $ is decreasing throughout real line
Show Answer
Answer:
Correct Answer: C
Solution:
$ f(x)=x^{3}-10x^{2}+200x-10 $
$ f’(x)=3x^{2}-20x+200 $
For increasing $ f’(x)>0 $
therefore $ 3x^{2}-20x+200>0 $
$ 3[ x^{2}-\frac{20}{3}x+\frac{200}{3}+\frac{100}{9}-\frac{100}{9} ]>0 $
$ \Rightarrow 3[ {{( x-\frac{10}{3} )}^{2}}+\frac{500}{9} ]>0 $
$ \Rightarrow 3{{( x-\frac{10}{3} )}^{2}}+\frac{500}{3}>0 $
Always increasing throughout real line.
BETA
