A) \[\left( \sqrt{2},\sqrt{10} \right)\]
B) \[\left[ \sqrt{2},\sqrt{10} \right)\]
C) \[\left( \sqrt{2},\sqrt{10} \right]\]
D) \[\left[ \sqrt{2},\sqrt{10} \right]\]
Correct Answer: D
Solution :
[d] Clearly, domain of the function is [2, 4]. Now, \[f'(x)=\frac{1}{\sqrt{x-2}}-\frac{1}{2\sqrt{4-x}}\] \[f'(x)=0\] or \[\sqrt{x-2}=2\sqrt{4-x}\] or \[x-2=16-4x\] or \[x=\frac{18}{5}\] Now, \[f(2)=\sqrt{2},f\left( \frac{18}{5} \right)=2\sqrt{\frac{18}{5}-2}+\sqrt{4-\frac{18}{5}}=\sqrt{10},\] \[f(4)=2\sqrt{2}\] Hence, range of the function is \[[\sqrt{2},\sqrt{10}]\]. Also, here \[x=(18/5)\] is the point of global maxima.
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