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Question: If \(f(x) = \frac{1}{\sqrt{x + 2\sqrt{2x - 4}}} + \frac{1}{\sqrt{x - 2\sqrt{2x - 4}}}\) for \(x > 2,...

If f(x)=1x+22x−4+1x−22x−4f(x) = \frac{1}{\sqrt{x + 2\sqrt{2x - 4}}} + \frac{1}{\sqrt{x - 2\sqrt{2x - 4}}} for x>2,x > 2, then

f(11)=f(11) =

A

76\frac{7}{6}

B

56\frac{5}{6}

C

67\frac{6}{7}

D

57\frac{5}{7}

Answer

67\frac{6}{7}

Explanation

Solution

f(x)=1x+22x−4+1x−22x−4f(x) = \frac{1}{\sqrt{x + 2\sqrt{2x - 4}}} + \frac{1}{\sqrt{x - 2\sqrt{2x - 4}}}

f(11)=111+218+111−218=13+2+13−2=3−27+3+27=67f(11) = \frac{1}{\sqrt{11 + 2\sqrt{18}}} + \frac{1}{\sqrt{11 - 2\sqrt{18}}} = \frac{1}{3 + \sqrt{2}} + \frac{1}{3 - \sqrt{2}} = \frac{3 - \sqrt{2}}{7} + \frac{3 + \sqrt{2}}{7} = \frac{6}{7}.