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Let f(x)=2cos e c2x+secx+cos e cxdot The...

Let `f(x)=2cos e c2x+secx+cos e cxdot` Then find the minimum value of `f(x)forx in (0,pi/2)dot`

Text Solution

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`f(x)=2co sec 2 x +sec x +co secx`
`=2 co sec 2x+(sqrt(secx)-sqrt(co sec x))^(2)+(2sqrt(2))/(sqrt(2cos x.sinx))`
`therefore =2 co sec 2x+(sqrt(secx)-sqrt(co sec x))^(2)+2sqrt(2)sqrt(co sec 2x)`
Clearly, minimum value occurs for `x=(pi)/(4)`, as for this value of x,
`sqrt(sec x)=sqrt(co secx)=0` and `co sec 2x=1`.
So, `f_("min"(x=(pi)/(4))=2(sqrt(2)+1)`
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