A) \[0<x<1\]
B) \[-1<x<1\]
C) \[x<-1\]or\[x>1\]
D) \[-1<x<-\frac{1}{2}\]
Correct Answer: C
Solution :
Key Idea: For increasing function,\[f'(x)>0\]and decreasing function,\[f'(x)<0\]. Let \[f(x)=2{{x}^{3}}-6x+5\] On differentiating, we get \[f'(x)=6{{x}^{2}}-6\] For increasing function,\[f'(x)>0\] \[\Rightarrow \] \[6{{x}^{2}}-6>0\Rightarrow {{x}^{2}}>1\] \[\Rightarrow \] \[x<-1\]or\[x>1\] Note: (i) For increasing function, if \[{{x}_{1}}>{{x}_{2}}\]or\[{{x}_{1}}<{{x}_{2}}\] \[\Rightarrow \] \[f({{x}_{1}})>f({{x}_{2}})\]or\[f({{x}_{1}})<f({{x}_{2}})\] (ii) For decreasing function, if \[{{x}_{1}}>{{x}_{2}}\]or\[{{x}_{1}}<{{x}_{2}}\] \[\Rightarrow \] \[f({{x}_{1}})<f({{x}_{2}})\]or\[f({{x}_{1}})>f({{x}_{2}})\]
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