Range of function $y=x^2 + \frac{4}{x^2+9}$ 
Please don't give the solution, I already got the answer by a different method.
I want to know why the method in the picture is wrong? Why cannot we simple add inequalities like that to get the interval of range and then find the minimum value from that range?
The correct answer to this question is : 4/9 ( minimum value)


You're right that $t\ge9$ and $0<4/t\le 4/9$. Therefore
$$
t+\frac{4}{t}-9\ge0
$$
However, this does not tell you that the minimum is $0$, which is not even an attained value. Indeed, you have $t\ge9$ and $4/t>0$, so certainly
$$
t+\frac{4}{t}\color{red}{>}9
$$
Thus, just considering those inequalities is not sufficient for determining the minimum value.

