How to find global maxima in open intervals

Question

The meaning of global maxima of a function $f(x)$ in $(a,b)$ is the highest value obtained by the function $f(x)$ in $(a,b).$
For example lets take the function $f(x) = 2x^3 - 9x^2+12x+6$. What would be the maxima of this function in the range $(0,6).$
Finding roots of derivative gives us local maxima at $x=1$ and $f(1) = 11$.
But clearly there are other values in of $x \in (0,6)$ where $f(x) > f(1)$. So how can we find the global maxima?

Answer 1

On a closed interval, a continuously differentiable function can achieve its maximum either at a local maximum or at either of the edges of the interval.

On an open interval, a continuously differentiable function can either achieve its maximum at a local maximum, or it may not have a maximum at all. For exmaple, the function $f(x)=x$ has a supremum of $1$ on $(0,1)$, but it has no maximum.

Answer 2

The function is increasing in $(0,1)$ and decreasing in $(1,2)$ and increasing in $(2,6)$. Its supremum is the maximum of $f(1)$ and $f(6)$ (which is $f(6)$) but this supremum is not attained in the open interval $(0,6)$.