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Question

Let f(x)=4excos2x9excot2x,g(x)=4ex+9ex, then the minimum value of g(x)f(x) is

A
6
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B
12
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C
36
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D
does not exist.
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Solution

The correct option is B 12
Given : f(x)=4excos2x9excot2x
f(x)=4ex(1sin2x)9ex(cosec2x1)f(x)=(4ex+9ex)(4exsin2x+9excosec2x) and g(x)=4ex+9ex
g(x)f(x)=4exsin2x+9excosec2x
As all terms are non-negative : using A.M.G.M.

4exsin2x+9excosec2x24exsin2x9excosec2x4exsin2x+9excosec2x2×6g(x)f(x)12



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