Trig Expression Minimum Value?

Question

Find the minimum value of

216((xsinx)^2)/5 + 36xsinx + 30 + 25/(xsinx) + 125/(6(xsinx)^2)

for 0 < x < pi.

Wow! Really nice work all around to find that factorization after differentiating. I also noted that the terms form a geometric progression but wasn't sure if it was worth pursuing. I'll have to mull over which to pick as best answer. There's also a non-calculus solution which I'll post tomorrow.

Alternate Solution:

Let y = xsinx. Then the expression becomes

S = 216(y^2)/5 + 36y + 30 + 25/y + 125/(6y^2).

Since 0
[216(y^2)/5 + 36y + 30 + 25/y + 125/(6y^2)]/5 = S/5 >= [(216y^2)/5 * 36y * 30 * 25/y * 125/(6y^2)] ^ (1/5).

The right hand side of the inequality simplifies to 30 giving us

S/5 >= 30
S >= 150.

Equality occurs if the terms in the expression (S) are all equal. This happens when y = 5/6. Thus, the minimum value of the expression is 150, and it occurs when xsinx = 5/6. We could use the Intermediate Value Thm to show xsinx can equal 5/6 on 0

Answer 1

Scythian - I think you should check your working. Or plot the function, at least. Note that at x = 1.306373 I get f(x) = 177.013.

Phineas' approach looks like a good one. To carry it further:
We want to minimise f(y) = 1/30[(36y + 15 + 25/y)^2 - 1125].
df/dy = (1/30) (2) (36y + 15 + 25/y) (36 - 25/y^2)
As Phineas noted, for x in (0, &pi) we have y > 0 and hence 36y + 15 + 25/y > 0. So for critical points we have y = ±5/6. Obviously only y = +5/6 is valid in the interval given. From the first derivative test we can verify that this is a minimum for the function.

Then we have f(y) = 216 (25/36) / 5 + 36 (5/6) + 30 + 25 (6/5) + 125/6 (36/25)
= 30 + 30 + 30 + 30 + 30 = 150.

Fortunately we're not asked to find x - of course we could only do this numerically. I get solutions 0.9941076 and 2.8442403 (to 7 d.p.).

Answer 2

Let u = x*sinx

I'm trying to follow your parenthesis here. The function reduces to
f(u) = 216/5 u^2 + 36u + 30 + 25/u + 125/6u^2

Differentiating once and rearranging
u^3 * f'(u) = 432/5 * u^4 + 36u^3 - 25u - 125/3

The quartic polynomial on the RHS has 2 rational roots at
u = +/- 5/6

f(5/6) = 150
f(-5/6) = 0

A simple plot reveals that f(5/6) = 150 is the minimum we seek. We can even go on to show that f"(5/6) > 0, but that will not be necessary.

ANS: 150

There will actually be 2 minima in this range since xsinx = 5/6 has 2 solutions. The graph looks something like a W.

*EDIT*
Phineas, I got 120 at first because I forgot the constant 30 inthe original function. But I corrected that mistake.

Answer 3

What a mess. As a numerical approximation, the minimum occurs at x = 1.306373, where the function has the minimum value of 23466.518.

Addendum: Maybe I am somehow misreading the expression, but I keep getting around (216)(36+30+25) ≈ (216)(100) = 21600 as the minimum. I don't see how I can get very much less than 21600, nothing like the other answers here. The question asks for the minimum valule for 0 < x < π, not 0 < x < 2π, which would in fact yield a lower minimum, a negative value. What's going on?

Addendum 2: Okay, finally maybe I see it. The 216 factor is only part of the first term, not the entire expression. Hold on, then.

All right, the minimum is 150, at x = 2.84424.

Answer 4

I don't know if I can make it through this tonight, but I will say that I would suggest multiplying through by 30, to make all the coefficients integers. Then letting y = xsinx and trying to complete the square to hopefully get something of the form:
(ay +b + c/y)^2 + d...

If we let y=xsinx and multiply through by 30, we get:

1296y^2 + 1080y + 900 + 750/y + 625/y^2 =
(36y + 15 + 25/y)^2 - 1125 =

Well I can say that -1125 is certainly a lower bound for the minimum, since squares are nonnegative, but it remains to be seen whether 36y + 15 + 25/y equals 0 for some y.

It alas does not since xsinx is positive between 0 and pi. However, we now have a much easier problem, of minimizing 36y + 25/y = 36xsinx - 25/xsinx.

Nice work, Scarlet Manuka!!! Incidentally, your background is much like mine, if you replace math with cs, physics with math, reading with tv, and 3 with 1. Then again with all of those replacements maybe it isn't so similar =)

Hmmm... DrD got what appears to be a different answer....
Now it matches -- cool!