Generalizing $N$ for different cases of a functions maxima.
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Given a number $N geq 5$. We have to divide it into equal parts so that the product of these parts is maximized. So using this I get
$$f(x) = left(dfracNxright)^x$$
I differentiate this to get $x$ as $N/e$ as the point of maxima.
However note that here $x$ is the "no of parts, and thus it has to be an integer, i.e., $x in mathbb Z_+$.
There are four options (more than one may be correct)
- $leftlfloor dfracNe rightrfloor$
- $leftlfloor dfracNe rightrfloor + 1$
- $leftlfloor dfracNe^2 rightrfloor$
- $leftlfloor dfracNe^2 rightrfloor + 1$
So eliminating the last two. we are left with two options. I did a little bit of hit and trial and found out that both the options may be correct according to the value of $N$. However, I am unable to find out a general expression as to what values of $N$ will give a maximum with the second option and which will give it with the first.
If someone could help me with this, please?
And also the fact as to why $N$ is greater than or equal to $5$. Maybe, because with $N =4$, both the first and the second options give the same answer? That's just a guess though.
calculus derivatives maxima-minima
edited Sep 3 at 12:55
Nash J.
1,104315
asked Sep 3 at 12:23
Pranav
207
add a comment |Â
up vote
2
down vote
favorite
Given a number $N geq 5$. We have to divide it into equal parts so that the product of these parts is maximized. So using this I get
$$f(x) = left(dfracNxright)^x$$
I differentiate this to get $x$ as $N/e$ as the point of maxima.
However note that here $x$ is the "no of parts, and thus it has to be an integer, i.e., $x in mathbb Z_+$.
There are four options (more than one may be correct)
- $leftlfloor dfracNe rightrfloor$
- $leftlfloor dfracNe rightrfloor + 1$
- $leftlfloor dfracNe^2 rightrfloor$
- $leftlfloor dfracNe^2 rightrfloor + 1$
So eliminating the last two. we are left with two options. I did a little bit of hit and trial and found out that both the options may be correct according to the value of $N$. However, I am unable to find out a general expression as to what values of $N$ will give a maximum with the second option and which will give it with the first.
If someone could help me with this, please?
And also the fact as to why $N$ is greater than or equal to $5$. Maybe, because with $N =4$, both the first and the second options give the same answer? That's just a guess though.
calculus derivatives maxima-minima
edited Sep 3 at 12:55
Nash J.
1,104315
asked Sep 3 at 12:23
Pranav
207
Your question is a bit tricky to read. Please rephrase it, use proper grammar and also use MathJax to format the equations. Then it's more likely that people will read your question and help you.
– Matti P.
Sep 3 at 12:31
Also, I think you are already practically at the solution. You just need to plug in different values and you'll hopefully understand what's going on.
– Matti P.
Sep 3 at 12:40
I wrote a code in C++ that told me for each corresponding value of N which one of these options will be max. However i was unable to find a pattern in the first 50 or so values
– Pranav
Sep 3 at 12:57
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Given a number $N geq 5$. We have to divide it into equal parts so that the product of these parts is maximized. So using this I get
$$f(x) = left(dfracNxright)^x$$
I differentiate this to get $x$ as $N/e$ as the point of maxima.
However note that here $x$ is the "no of parts, and thus it has to be an integer, i.e., $x in mathbb Z_+$.
There are four options (more than one may be correct)
- $leftlfloor dfracNe rightrfloor$
- $leftlfloor dfracNe rightrfloor + 1$
- $leftlfloor dfracNe^2 rightrfloor$
- $leftlfloor dfracNe^2 rightrfloor + 1$
So eliminating the last two. we are left with two options. I did a little bit of hit and trial and found out that both the options may be correct according to the value of $N$. However, I am unable to find out a general expression as to what values of $N$ will give a maximum with the second option and which will give it with the first.
If someone could help me with this, please?
And also the fact as to why $N$ is greater than or equal to $5$. Maybe, because with $N =4$, both the first and the second options give the same answer? That's just a guess though.
calculus derivatives maxima-minima
edited Sep 3 at 12:55
Nash J.
1,104315
asked Sep 3 at 12:23
Pranav
207
Given a number $N geq 5$. We have to divide it into equal parts so that the product of these parts is maximized. So using this I get
$$f(x) = left(dfracNxright)^x$$
I differentiate this to get $x$ as $N/e$ as the point of maxima.
However note that here $x$ is the "no of parts, and thus it has to be an integer, i.e., $x in mathbb Z_+$.
There are four options (more than one may be correct)
- $leftlfloor dfracNe rightrfloor$
- $leftlfloor dfracNe rightrfloor + 1$
- $leftlfloor dfracNe^2 rightrfloor$
- $leftlfloor dfracNe^2 rightrfloor + 1$
So eliminating the last two. we are left with two options. I did a little bit of hit and trial and found out that both the options may be correct according to the value of $N$. However, I am unable to find out a general expression as to what values of $N$ will give a maximum with the second option and which will give it with the first.
If someone could help me with this, please?
And also the fact as to why $N$ is greater than or equal to $5$. Maybe, because with $N =4$, both the first and the second options give the same answer? That's just a guess though.
calculus derivatives maxima-minima
calculus derivatives maxima-minima
edited Sep 3 at 12:55
Nash J.
1,104315
asked Sep 3 at 12:23
Pranav
207
edited Sep 3 at 12:55
Nash J.
1,104315
asked Sep 3 at 12:23
Pranav
207
edited Sep 3 at 12:55
Nash J.
1,104315
edited Sep 3 at 12:55
Nash J.
1,104315
edited Sep 3 at 12:55
Nash J.
1,104315
1,104315
asked Sep 3 at 12:23
Pranav
207
asked Sep 3 at 12:23
Pranav
207
asked Sep 3 at 12:23
Pranav
207
207
Your question is a bit tricky to read. Please rephrase it, use proper grammar and also use MathJax to format the equations. Then it's more likely that people will read your question and help you.
– Matti P.
Sep 3 at 12:31
Also, I think you are already practically at the solution. You just need to plug in different values and you'll hopefully understand what's going on.
– Matti P.
Sep 3 at 12:40
I wrote a code in C++ that told me for each corresponding value of N which one of these options will be max. However i was unable to find a pattern in the first 50 or so values
– Pranav
Sep 3 at 12:57
add a comment |Â
Your question is a bit tricky to read. Please rephrase it, use proper grammar and also use MathJax to format the equations. Then it's more likely that people will read your question and help you.
– Matti P.
Sep 3 at 12:31
Also, I think you are already practically at the solution. You just need to plug in different values and you'll hopefully understand what's going on.
– Matti P.
Sep 3 at 12:40
I wrote a code in C++ that told me for each corresponding value of N which one of these options will be max. However i was unable to find a pattern in the first 50 or so values
– Pranav
Sep 3 at 12:57
Your question is a bit tricky to read. Please rephrase it, use proper grammar and also use MathJax to format the equations. Then it's more likely that people will read your question and help you.
– Matti P.
Sep 3 at 12:31
Your question is a bit tricky to read. Please rephrase it, use proper grammar and also use MathJax to format the equations. Then it's more likely that people will read your question and help you.
– Matti P.
Sep 3 at 12:31
Also, I think you are already practically at the solution. You just need to plug in different values and you'll hopefully understand what's going on.
– Matti P.
Sep 3 at 12:40
Also, I think you are already practically at the solution. You just need to plug in different values and you'll hopefully understand what's going on.
– Matti P.
Sep 3 at 12:40
I wrote a code in C++ that told me for each corresponding value of N which one of these options will be max. However i was unable to find a pattern in the first 50 or so values
– Pranav
Sep 3 at 12:57
I wrote a code in C++ that told me for each corresponding value of N which one of these options will be max. However i was unable to find a pattern in the first 50 or so values
– Pranav
Sep 3 at 12:57
add a comment |Â
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Your question is a bit tricky to read. Please rephrase it, use proper grammar and also use MathJax to format the equations. Then it's more likely that people will read your question and help you.
– Matti P.
Sep 3 at 12:31
Also, I think you are already practically at the solution. You just need to plug in different values and you'll hopefully understand what's going on.
– Matti P.
Sep 3 at 12:40
I wrote a code in C++ that told me for each corresponding value of N which one of these options will be max. However i was unable to find a pattern in the first 50 or so values
– Pranav
Sep 3 at 12:57