Shown below is a cylinder shaped wine barrel.

Fig. 1

From Fig. 1, we see that

and so,

Kepler’s “Wine Barrel Problem” can be stated as:

If is fixed, what value of gives the largest volume of ?

Kepler conducted extensive numerical studies on this problem. However, it was solved analytically only after the invention of calculus.

In the spring of 2012, while carrying out a research on solving maximization/minimization problems, I discovered the following theorem:

**Theorem-1**. For positive quantities and positive rational quantities , if is a constant, then attains its maximum if .

By applying Theorem-1, the “Wine Barrel Problem” can be solved analytically without calculus at all. It is as follows:

Rewrite (2) as

Since

and , a constant,

by Theorem-1, when

or

(see (3)) attains its maximum.

Solving (4) for positive , we have

Discovered from the same research is another theorem for solving minimization problem without calculus:

**Theorem-2**. For positive quantities and positive rational quantities , if is a constant, then attains its minimum if .

Let’s look at an example:

*Problem: Find the minimum value of for .*

Since for and , a constant,

by Theorem-2, when

attains its minimum.

Solving (5) for yields

.

Therefore, at attains its minimum value (see Fig. 2).

Fig. 2

Nonetheless, neither Theorem-1 nor Theorem-2 is a silver bullet for solving max/min problems without calculus. For example,

*Problem: Find the minimum value of for .*

Theorem-2 is not applicable here (see *Exercise-1)*. To solve this problem, we proceed as follows:

From , we have

and so,

.

That is,

.

Or,

i.e.

.

Since ,

,

with the “=” sign in “” holds at .

Therefore, attains its minimum -54 at (see FIg. 3).

Fig. 3

*Exercise-1* Explain why Theorem-2 is not applicable to finding the minimum of for .

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