$f(x)=max_0le yle 1fracx+y+1$ [closed]

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Let $f:[0,1]tomathbb R$ be defined as
$$f(x)=maxleftfracx-yx+y+1: 0le yle 1righttext for 0le xle 1$$
Then which of the following statements is correct?

$f$ is strictly increasing on $[0,1/2]$ and strictly decreasing on $[1/2,1]$

$f$ is strictly decreasing on $[0,1/2]$ and strictly increasing on $[1/2,1]$

$f$ is strictly increasing on $[0,(sqrt3-1)/2]$ and strictly decreasing on $[(sqrt3-1)/2,1]$

$f$ is strictly decreasing on $[0,(sqrt3-1)/2]$ and strictly increasing on $[(sqrt3-1)/2,1]$

edited Jul 21 at 10:07

Lorenzo B.

1,5402418

asked Jul 21 at 5:02

Alphanerd

6317

closed as off-topic by uniquesolution, Martin R, Tyrone, amWhy, Adrian Keister Jul 24 at 13:23

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â€“ uniquesolution, Martin R, Tyrone, amWhy, Adrian Keister

If this question can be reworded to fit the rules in the help center, please edit the question.

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up vote
-1
down vote

favorite

Let $f:[0,1]tomathbb R$ be defined as
$$f(x)=maxleftfracx-yx+y+1: 0le yle 1righttext for 0le xle 1$$
Then which of the following statements is correct?

$f$ is strictly increasing on $[0,1/2]$ and strictly decreasing on $[1/2,1]$

$f$ is strictly decreasing on $[0,1/2]$ and strictly increasing on $[1/2,1]$

$f$ is strictly increasing on $[0,(sqrt3-1)/2]$ and strictly decreasing on $[(sqrt3-1)/2,1]$

$f$ is strictly decreasing on $[0,(sqrt3-1)/2]$ and strictly increasing on $[(sqrt3-1)/2,1]$

edited Jul 21 at 10:07

Lorenzo B.

1,5402418

asked Jul 21 at 5:02

Alphanerd

6317

closed as off-topic by uniquesolution, Martin R, Tyrone, amWhy, Adrian Keister Jul 24 at 13:23

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â€“ uniquesolution, Martin R, Tyrone, amWhy, Adrian Keister

If this question can be reworded to fit the rules in the help center, please edit the question.

add a commentÂ |Â

up vote
-1
down vote

favorite

Let $f:[0,1]tomathbb R$ be defined as
$$f(x)=maxleftfracx-yx+y+1: 0le yle 1righttext for 0le xle 1$$
Then which of the following statements is correct?

$f$ is strictly increasing on $[0,1/2]$ and strictly decreasing on $[1/2,1]$

$f$ is strictly decreasing on $[0,1/2]$ and strictly increasing on $[1/2,1]$

$f$ is strictly increasing on $[0,(sqrt3-1)/2]$ and strictly decreasing on $[(sqrt3-1)/2,1]$

$f$ is strictly decreasing on $[0,(sqrt3-1)/2]$ and strictly increasing on $[(sqrt3-1)/2,1]$

edited Jul 21 at 10:07

Lorenzo B.

1,5402418

asked Jul 21 at 5:02

Alphanerd

6317

Let $f:[0,1]tomathbb R$ be defined as
$$f(x)=maxleftfracx-yx+y+1: 0le yle 1righttext for 0le xle 1$$
Then which of the following statements is correct?

$f$ is strictly increasing on $[0,1/2]$ and strictly decreasing on $[1/2,1]$

$f$ is strictly decreasing on $[0,1/2]$ and strictly increasing on $[1/2,1]$

$f$ is strictly increasing on $[0,(sqrt3-1)/2]$ and strictly decreasing on $[(sqrt3-1)/2,1]$

$f$ is strictly decreasing on $[0,(sqrt3-1)/2]$ and strictly increasing on $[(sqrt3-1)/2,1]$

edited Jul 21 at 10:07

Lorenzo B.

1,5402418

asked Jul 21 at 5:02

Alphanerd

6317

edited Jul 21 at 10:07

Lorenzo B.

1,5402418

edited Jul 21 at 10:07

Lorenzo B.

1,5402418

edited Jul 21 at 10:07

Lorenzo B.

1,5402418

asked Jul 21 at 5:02

Alphanerd

6317

asked Jul 21 at 5:02

Alphanerd

6317

asked Jul 21 at 5:02

Alphanerd

6317

closed as off-topic by uniquesolution, Martin R, Tyrone, amWhy, Adrian Keister Jul 24 at 13:23

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â€“ uniquesolution, Martin R, Tyrone, amWhy, Adrian Keister

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by uniquesolution, Martin R, Tyrone, amWhy, Adrian Keister Jul 24 at 13:23

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â€“ uniquesolution, Martin R, Tyrone, amWhy, Adrian Keister

If this question can be reworded to fit the rules in the help center, please edit the question.

add a commentÂ |Â

1 Answer
1

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up vote
2
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accepted

This problem requires more work than you expected!

Extremal problems with parameters are tricky, since not only the value of the maximum, but also the "morphology of the extremal configuration" depends on the parameter. In the case at hand I propose to introduce a function $g$ of two variables as follows:
$$g_x(y):=left{eqalignx-yover x+y+1qquad&(0leq yleq x)cr
y-xover x+y+1qquad&(xleq yleq 1)crright.$$
Now consider $xin[0,1]$ as fixed. Then analyze the graph of the function $ymapsto g_x(y)$ in the $y$-interval $[0,x]$, where $g_x(y)=x-yover x+y+1$, and similarly analyze the graph of the function $ymapsto g_x(y)$ in the $y$-interval $[x,1]$, where $g_x(y)=x-yover x+y+1$. Now you have a complete overview over the function $ymapsto g_x(y)$ for this particular $x$. There will be a maximum value, and this is your $f(x)$ for that particular $x$.

Note that $ymapsto g_x(y)$ is monotone on the subintervals $[0,x]$ and $[x,1]$. It follows that
$$f(x)=maxbiglg_x(0), g_x(x),g_x(1)bigr=maxleftxover x+1, 0,1-xover x+2right .$$

edited Jul 21 at 11:57

answered Jul 21 at 10:03

Christian Blatter

163k7107306

This problem appeared in a test conducted by ISI(Indian Statistical Institute) which is given by high school students. I am a bit surprised that they gave such a complex question and looking at your solution , the question seems to be a bit out of scope . Anyways , thank you for your solution . I will try my best to grasp it .
â€“Â Alphanerd
Jul 21 at 10:26

add a commentÂ |Â

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

up vote
2
down vote

accepted

This problem requires more work than you expected!

Note that $ymapsto g_x(y)$ is monotone on the subintervals $[0,x]$ and $[x,1]$. It follows that
$$f(x)=maxbiglg_x(0), g_x(x),g_x(1)bigr=maxleftxover x+1, 0,1-xover x+2right .$$

edited Jul 21 at 11:57

answered Jul 21 at 10:03

Christian Blatter

163k7107306

This problem appeared in a test conducted by ISI(Indian Statistical Institute) which is given by high school students. I am a bit surprised that they gave such a complex question and looking at your solution , the question seems to be a bit out of scope . Anyways , thank you for your solution . I will try my best to grasp it .
â€“Â Alphanerd
Jul 21 at 10:26

add a commentÂ |Â

up vote
2
down vote

accepted

This problem requires more work than you expected!

Note that $ymapsto g_x(y)$ is monotone on the subintervals $[0,x]$ and $[x,1]$. It follows that
$$f(x)=maxbiglg_x(0), g_x(x),g_x(1)bigr=maxleftxover x+1, 0,1-xover x+2right .$$

edited Jul 21 at 11:57

answered Jul 21 at 10:03

Christian Blatter

163k7107306

This problem appeared in a test conducted by ISI(Indian Statistical Institute) which is given by high school students. I am a bit surprised that they gave such a complex question and looking at your solution , the question seems to be a bit out of scope . Anyways , thank you for your solution . I will try my best to grasp it .
â€“Â Alphanerd
Jul 21 at 10:26

add a commentÂ |Â

up vote
2
down vote

accepted

This problem requires more work than you expected!

Note that $ymapsto g_x(y)$ is monotone on the subintervals $[0,x]$ and $[x,1]$. It follows that
$$f(x)=maxbiglg_x(0), g_x(x),g_x(1)bigr=maxleftxover x+1, 0,1-xover x+2right .$$

edited Jul 21 at 11:57

answered Jul 21 at 10:03

Christian Blatter

163k7107306

This problem requires more work than you expected!

Note that $ymapsto g_x(y)$ is monotone on the subintervals $[0,x]$ and $[x,1]$. It follows that
$$f(x)=maxbiglg_x(0), g_x(x),g_x(1)bigr=maxleftxover x+1, 0,1-xover x+2right .$$

edited Jul 21 at 11:57

answered Jul 21 at 10:03

Christian Blatter

163k7107306

edited Jul 21 at 11:57

answered Jul 21 at 10:03

Christian Blatter

163k7107306

answered Jul 21 at 10:03

Christian Blatter

163k7107306

answered Jul 21 at 10:03

Christian Blatter

163k7107306

This problem appeared in a test conducted by ISI(Indian Statistical Institute) which is given by high school students. I am a bit surprised that they gave such a complex question and looking at your solution , the question seems to be a bit out of scope . Anyways , thank you for your solution . I will try my best to grasp it .
â€“Â Alphanerd
Jul 21 at 10:26

add a commentÂ |Â

This problem appeared in a test conducted by ISI(Indian Statistical Institute) which is given by high school students. I am a bit surprised that they gave such a complex question and looking at your solution , the question seems to be a bit out of scope . Anyways , thank you for your solution . I will try my best to grasp it .
â€“Â Alphanerd
Jul 21 at 10:26

This problem appeared in a test conducted by ISI(Indian Statistical Institute) which is given by high school students. I am a bit surprised that they gave such a complex question and looking at your solution , the question seems to be a bit out of scope . Anyways , thank you for your solution . I will try my best to grasp it .
â€“Â Alphanerd
Jul 21 at 10:26

add a commentÂ |Â

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