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
Correct Answer: C
Solution :
\[f(x)={{x}^{3}}-10{{x}^{2}}+200x-10\] \[f'(x)=3{{x}^{2}}-20x+200\] For increasing \[f'(x)>0\] Þ \[3{{x}^{2}}-20x+200>0\] \[3\text{ }\left[ {{x}^{2}}-\frac{20}{3}x+\frac{200}{3}+\frac{100}{9}-\frac{100}{9} \right]>0\] \[\Rightarrow 3\text{ }\left[ {{\left( x-\frac{10}{3} \right)}^{2}}+\frac{500}{9} \right]>0\] \[\Rightarrow 3\text{ }{{\left( x-\frac{10}{3} \right)}^{2}}+\frac{500}{3}>0\] Always increasing throughout real line.
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