Find the minimum value of this function for $0\lt \theta \lt \pi$ $$y = 3\sin{\theta} + \text{cosec}^3\theta$$
What I want to specifically ask is about use of inequality rules in this function.
I used the rule: $$\text{if } a\lt b \text{ and } c\lt d \implies a + c\lt b + d$$
$$0\lt3\sin\theta\lt 3\text{ }\forall\text{ } 0\lt \theta \lt \pi$$
Also $$1\lt\text{cosec}^3\theta\lt \infty$$
So shouldn't these two inequations combine to give $$1\lt3\sin{\theta} + \text{cosec}^3\theta\lt\infty$$
But the correct answer is 4. Did I do any mistake or does this rule not apply here?
I'm not looking for solution to this problem, I just want to know why this method fails here.
Best Answer
The property $y>1$ is true, but it does not give a minimal value because you need also to show that the minimal value occurs, as in the following reasoning.
By AM-GM $$y\geq4\sqrt[4]{\sin^3x\cdot\frac{1}{\sin^3x}}=4.$$ The equality occurs for $x=\frac{\pi}{2},$ which says that we got a minimal value.