A) \[x=0\]is a maximum
B) \[x=0\]is a minimum
C) \[x=-1\] is a maximum
D) \[x=-1\] is a minimum
Correct Answer: D
Solution :
Given, \[f(x)=x{{e}^{x}}\] \[\Rightarrow \] \[f'(x)={{e}^{x}}+x{{e}^{x}}\] \[\Rightarrow \] \[f''(x)={{e}^{x}}+x{{e}^{x}}+{{e}^{x}}=2{{e}^{x}}+x{{e}^{x}}\] For maxima or minima, put \[f'(x)=0\] \[\Rightarrow \] \[{{e}^{x}}(1+x)=0\] \[\Rightarrow \] \[x=-1\] At \[x=-1,\,\,f''(x)>0\] \[\therefore \] At \[x=-1,\,\,f(x)\] is minimum.
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