# 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?

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

• can you elaborate how did you use AM-GM inequality?
– D S
Commented Dec 28, 2022 at 4:41
• @DS (25 cosec²x + 16 sin²x)/2 >= √(25*16) sorry if u can't understand I don't know how to write you use special keys to write maths . Commented Dec 31, 2022 at 13:51

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:

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.

• That's it. now can you tell me that how solution for this ( when csc² coefficient is larger) is given by book a²+b²...( Sum of squares of coefficient) Commented Dec 29, 2022 at 2:45
• Because of the AM-GM inequality and the fact that $\csc \theta \sin \theta = 1$. It's just that, in the first case (in the graph above), the coefficients were not large enough to have the ranges/graphs intersect. Commented Dec 29, 2022 at 2:51
• Then how to come to a²+b² answer. Commented Jun 19, 2023 at 17:36