Hello!
The AM-GM inequality states that for any non-negative numbers `a` and `b`
`(a+b)/2gt=sqrt(a*b),` or `a+bgt=2sqrt(a*b).`
Therefore it may be suitable for estimating a sum from the below.
Denote `sin^2(x)=u,` then `cos^2(x)=1-u` and the function becomes
`3^(u^3)+3^((1-u)^3).`
`u` may be any number in `[0,1].`
We may apply this for our function and obtain an inequality
`3^(u^3)+3^((1-u)^3) gt= 2sqrt(3^(u^3)*3^((1-u)^3)) =`
`= 2*3^(1/2(u^3+(1-u)^3)) = 2*3^(1/2(1-3u+3u^2)).`
The exponent `1/2(1-3u+3u^2)` has one and only one minimum at `u_0=1/2 in [0.1],` the value at `u_0` is `1/8.`
So for any `u` we have `3^(u^3)+3^((1-u)^3) gt= 2*3^(1/8),` and this inequality becomes an equality only at `u_0=1/2.`
Now recall that `u=sin^2(x).` It is equal to `1/2` when `sin(x)=+-1/sqrt(2),` so at `x=pi/4+(k pi)/2.` This way we have found minimums of the given function. They are `pi/4+(k pi)/2,` and the minimum value is `2*3^(1/8) approx 2.294.`
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