A) \[\left[ \frac{2}{3},2 \right]\]
B) \[\left[ 0,\frac{2}{3} \right]\]
C) \[\left[ 0,2 \right]\]
D) \[\left[ -\frac{1}{3},\frac{2}{3} \right]\]
Correct Answer: A
Solution :
| we have \[y+z=4-x\]and \[{{y}^{2}}+{{z}^{2}}=6-{{x}^{2}}.\] |
| Also, \[yz=\frac{1}{2}[{{(y+z)}^{2}}-({{y}^{2}}+{{z}^{2}})]={{x}^{2}}-4x+5\] |
| Therefore y, z must be roots of the |
| equation \[{{t}^{2}}-(4-x)t+{{x}^{2}}-4x+5=0.\] |
| As y and z are real, so \[{{(4-x)}^{2}}-4({{x}^{2}}-4x+5)\ge 0\Rightarrow \frac{2}{3}\le x\le 2\] |
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