The Minimum value of `27^cosx +81^sinx` is equal to

Options

A. `(2)/(3sqrt3)`
B. `(2)/(9sqrt3)`
C. `(4)/(3sqrt3)`
D. none of these

Answer

B

Explanation

We known that
`A.M. ge G.M.`
`therefore(27^(cosx)x+81^(sinx))/(2)gesqrt(27^(cosx)xx81^(sinx))`
`implies27^(cosx)+81^(sinx)ge2sqrt(3^(3cosx+4sinx))`
`implies27^(cosx) +81^(sinx)ge2xxsqrt(3^(-5))`
`" "[because-5le3cosx+4sinxle5]`
`implies27^(cosx)+81^(sinx)ge(2)/(9sqrt3)`
Hence, the minimum value of `27^(cosx)+81^(sinx)is(2)/(9sqrt3).`