Why does the AM-GM inequality not show $25 \csc^2(\theta) +16 \sin^2(\theta)$ has a minimum of $41$ as the graph indicates

a.m.-g.m.-inequalityinequalitytrigonometry

Let's say we have to find range of $f(\theta) = 25 \csc^2(\theta) +16 \sin^2(\theta)$

If I use $AM \ge GM$

Then $f(\theta) \ge 40$

Which tells minimum value of $f(\theta)$ will be $40$

But I checked it on graphing calculator and it is showing $41$ will be minimum value

Then I tried for
$f(\theta) = 16 \csc^2(\theta) +25 \sin^2(\theta)$

Now using $AM \ge GM$

I am getting 40 as answer of minimum value of function and also checked on graphing calculator

Now my question is why my first question answer is wrong using $AM \ge GM$

I think it is something related to coefficient but I am not getting it how it is wrong

Best Answer

The simple reason is that it does not give a minimum; it gives a lower bound, but it's not necessarily the largest or best lower bound.

Equality is achieved in AM-GM when each number is equal, i.e.

$$\frac{x+y}{2} = \sqrt{xy} \iff x=y$$

Of course, $f(\theta) \ge 40$ and the existence of $\theta_0$ such that $f(\theta_0) = 40$ would validate that the minimum is $40$, but the question remains: is there ever an $\theta$ for which

$$25 \csc^2 \theta \stackrel{?}{=} 16 \sin^2 \theta$$

holds?

Try graphing it and you will see that no $\theta$ works:

enter image description here

This also explains why the case of reversed coefficients work: graph $16 \csc^2 \theta$ and $25 \sin^2 \theta$, and you will find that the graphs sometimes intersect, so the AM-GM inequality will perfectly and accurately describe the minimum of their sum.

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