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Find minium value of (9x^(2)sin^(2)x+4)/...

Find minium value of `(9x^(2)sin^(2)x+4)/(2sinx.x)` in the interval `0 le x le pi`.

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Verified by Experts

As `x in (0,pi)` it means `sin gt 0`
Also, `(9x^(2)sin^(2)x+4)/(2sinx.x)` can be rearranged as `9/2 x sinx+2/(x sinx)`
Product of these two terms is constant
Applying A.M. `ge` G.M.
`=(9/2x sinx+2/(xsinx))/(2) ge sqrt(9)`
`=9/2 x sinx +2/(x sinx) ge 6`
`"min"((9x^(2)sin^(2)x+4)/(2xsinx))=6`
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