A) \[P=[3,\,\,\infty ),\]\[Q\in (-\infty ,\,\,4]\]
B) \[P=[2,\,\,\infty ),\]\[Q\in [4,\,\,\infty )\]
C) \[P=(-\,\infty ,\,\,2],\]\[Q\in [4,\,\,\infty )\]
D) \[P=(-\,\infty ,\,\,2],\]\[Q\in (-\,\infty ,\,\,4]\]
Correct Answer: D
Solution :
| \[f\,(x)=4x-{{x}^{2}}\] |
| \[f'(x)=4-2x\] |
| \[\therefore \] \[f'(x)>0\] if \[x<2\] |
| And \[f'(x)<0\] if \[x>2\] |
| \[\therefore \] if \[x<2,\]then f(x) is increasing |
| if \[x>2,\] then f(x) is decreasing |
| \[f\,(2)=4\] |
| (i) If \[P=(-\,\infty ,\,\,2],\] then f(x) is one - one and range \[=(-\,\infty ,\,\,4]\] |
| \[\therefore \] f(x) is onto if \[Q=(-\,\infty ,\,\,4]\] |
| (ii) If\[P=[2,\,\,\infty ),\]then f(x) is one - one and range \[=(-\,\infty ,\,\,4]\] |
| \[\therefore \] f(x) is onto if \[Q=(-\,\infty ,\,\,4]\] |
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