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\(f''(x) = \dfrac{3x(x^2+27)}{(x^2-9)^3}\)

\(\text{a point }x \text{ is an inflection point of }f(x) \text{ if both of the following hold}\\ \text{1) }f''(x)=0 \\ \text{2) }sgn(f''(x-\epsilon)) \neq sgn(f''(x+\epsilon)),~\epsilon > 0 \text{ but small}\)

\(\text{It's pretty clear that the only real value for which }f''(x)=0 \text{ is }x=0\\ sgn(f''(-\epsilon)) = 1\\ sgn(f''(\epsilon))=-1\\ \text{So }x=0 \text{ is indeed an inflection point and the only one.}\)

The reason why i asked so many similar question is because I am preparing for a proof by induction exam, i have attempted all of these questions myself. i was unaware that i was supposed to post my own working outs.

but i will stop from now on

35/3  * 8/8 =  280/24

63/8  * 3/3 = 189/24

GCF of 280 ND 189 IS ?   

    Answer ' 7'

so use 7/24  x 7/24 m tiles         280/24 x 189/24 = 91.875 sq m

                                                               91.875m^2/ (7/24 * 7.24)m^2/tile = 1080 tiles   (40 x 27 tiles)

I am sorry Cphill, but that is incorrect  Thank you for trying though

from this equation, you can get x = x, and then 0 = 0, which means that x can be everything. 

please someone help me

I don't know if there is a shortcut to it, but computers can easily handle it. It is 2001-digit long. 
It begins with......10000431729.......and ends in....  5137781330 mod 1000 =330 

It's a bit more clever than that.

If you look at k^1000 % 1000 you'll see it's a periodic sequence with period 10

I'm no number theory guru but I think the secret lies in the common factors of k and 1000.

I'll post if I figure it out.

In the question it asks for a accetable range so there has to be one I believe, i have to make candels

The problem does not tell us how much time or how many candles we can make, so I guess theoretically the maximum is \(\infty\) and the minimum is 0. 

If there was a restraint of time or whatever, for the number of candles \(n\), the maximum profit is: {selling price} - {making/producing price} * n. The minimum is $0, just don't make any candles. 

I'll probably feed the data into a computer program to determine the answer.

- PM

This is a really tedious number theory problem. I would probably try to find a pattern in the mod 1000 of the powers of 2, 3, 4, ect. Remember that any multiple of 10 to the power of 1000 (\(10^{1000}\), \(20^{1000}\), ect.) mod 1000 is 0, which means it can be divided by 1000. Other than that, I think there is just pure computations for this problem.

They part im confused at is how would i write a linear program model and constraits with those values as well as how i would be able to graph the feasible region and maxiumum and minium profits

8-5y=-3y-31

Yea, this makes more sense because we know that a candle costs $1.60 each. We also know the the cost of the brick. 

Yes, they do give an answer, but it is totally wrong!!. The powers of 10 given by Wolfram are equivalent to the tower of 7s. You start from the right to the left like this: 7^7^7^7^7^823,543 =7^7^7^7^(3.75982352  E+695974). The power of the 4th 7 is more than 695,974-digit long!! And so on......etc.

Maybe this will help , this is the question before it with the answer

You’re instructed to prepare for a promotional candle sale you will use to test the popularity of the two redesigned candles. Wicks, labor, and other costs add about $1.60 to the cost of each candle.

Suggest a retail price for each candle that will cover costs and allow for some profit. Explain how you determined these prices. Your reasoning does not have to be all math related, but it should include specific values derived from your calculations.

 Answer:

cost of the brick is $4.48 + $1.60 = $6.08

50% markup is typical on things like decorative candles so a profit of $3.04

for a total retail price of $9.12 .

I'd then figure the sales tax for my area.  Where I live it 8.25%, so adjusted for an even dollar value. A retail price of $9.87 comes to $10 with tax

cost of the egyptian is $7.00 + $1.60 = 8.60$

markup of $4.30 gives a retail price of $12.90

same story with the sales tax.  I'd price it at $13.96 so tax brings it to $14

That's ok, I'll get someone else to answer this question if you need help. 

Also, you said, "Note* The price for the Brick candle is $10.00 with tax of 8.25% tax and without it %12.90 and the Egyptian candle is $14.00 with tax of 8.25% and without it $13.96." Is "without thing" the price or percent? I think you meant $12.90. 

Let's assume you meant, "Note* The price for the Brick candle is $10.00 with tax of 8.25% tax and without it $12.90." Without the 8.25% tax, the price is $12.90, which is more expensive than the price with tax, $10.

I am not really sure myself

I am not sure I understand your question. Could you please elaborate more or give more information? 

And this online high precision calculator website: 

But that is powers of TEN...  not powers of SEVEN. 

NO! That is NOT the answer. No calculator can calculate it!! According to WolframAlpha, this is its representation as a power of 10s:  

10^(10^(10^(10^(10^(10^5.842593328962333))))). However, they do give the last few digits as

.........1565172343.

First one

C(30, 3)    = 4060 different combinations

Second one

30 possibilities * 29possibilities * 28 possibilities * ......* 3 * 2 * 1    =   30!