Find the minimum value of $3\sin^2x+27\csc^2x$.
Using derivative, we get the answer is $30$.
Using AM-GM inequality or completing the whole square, we get the minimum is $18$.
The answer is $30$.
Why can't we use the AM-GM method or completing the whole square method here?
AM-GM method tells you that the equality only holds if $a=b$, i.e., in your case $$3\sin^2 x=27 \csc^2 x$$ But, this can never be true as thuis implies $$\sin^4 x=9$$ but we have $$-1\leq\sin x\leq1$$ Does that help?