The range of $ab$ if $|a| le 1$ and $a + b = 1$, $a,binmathbb R$? [closed]











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The range of $ab$ if $|a| le 1$ and $a + b = 1$, $a,binmathbb R$ ?






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closed as off-topic by Jimmy R., Hans Lundmark, KReiser, José Carlos Santos, Christopher Nov 20 at 13:25


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." – Jimmy R., Hans Lundmark, KReiser, José Carlos Santos, Christopher

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









  • 1




    What are your attempts?
    – Aniruddha Deshmukh
    Nov 20 at 4:27










  • Used AM GM inequality , got max value Right , which is 1/4. But minimum ??
    – Alvin Carter
    Nov 20 at 5:29















up vote
-2
down vote

favorite












The range of $ab$ if $|a| le 1$ and $a + b = 1$, $a,binmathbb R$ ?






inequality







share|cite|improve this question















closed as off-topic by Jimmy R., Hans Lundmark, KReiser, José Carlos Santos, Christopher Nov 20 at 13:25


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." – Jimmy R., Hans Lundmark, KReiser, José Carlos Santos, Christopher

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









  • 1




    What are your attempts?
    – Aniruddha Deshmukh
    Nov 20 at 4:27










  • Used AM GM inequality , got max value Right , which is 1/4. But minimum ??
    – Alvin Carter
    Nov 20 at 5:29













up vote
-2
down vote

favorite









up vote
-2
down vote

favorite











The range of $ab$ if $|a| le 1$ and $a + b = 1$, $a,binmathbb R$ ?






inequality







share|cite|improve this question















The range of $ab$ if $|a| le 1$ and $a + b = 1$, $a,binmathbb R$ ?






inequality




inequality






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 20 at 4:29









Jimmy R.

33k42157




33k42157










asked Nov 20 at 4:23









Alvin Carter

13




13




closed as off-topic by Jimmy R., Hans Lundmark, KReiser, José Carlos Santos, Christopher Nov 20 at 13:25


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." – Jimmy R., Hans Lundmark, KReiser, José Carlos Santos, Christopher

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




closed as off-topic by Jimmy R., Hans Lundmark, KReiser, José Carlos Santos, Christopher Nov 20 at 13:25


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." – Jimmy R., Hans Lundmark, KReiser, José Carlos Santos, Christopher

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








  • 1




    What are your attempts?
    – Aniruddha Deshmukh
    Nov 20 at 4:27










  • Used AM GM inequality , got max value Right , which is 1/4. But minimum ??
    – Alvin Carter
    Nov 20 at 5:29














  • 1




    What are your attempts?
    – Aniruddha Deshmukh
    Nov 20 at 4:27










  • Used AM GM inequality , got max value Right , which is 1/4. But minimum ??
    – Alvin Carter
    Nov 20 at 5:29








1




1




What are your attempts?
– Aniruddha Deshmukh
Nov 20 at 4:27




What are your attempts?
– Aniruddha Deshmukh
Nov 20 at 4:27












Used AM GM inequality , got max value Right , which is 1/4. But minimum ??
– Alvin Carter
Nov 20 at 5:29




Used AM GM inequality , got max value Right , which is 1/4. But minimum ??
– Alvin Carter
Nov 20 at 5:29










1 Answer
1






active

oldest

votes

















up vote
1
down vote



accepted










Hint: $|a|ge 1$ implies that $-1le ale 1$. Also, $a+b=1$ implies that $b=1-a$, hence $$ab=a(1-a)$$



So, your task is to find the minimum and the maximum of the function $f(a)=a(1-a)$ in the interval $ain [-1,1]$. Can you do it?






share|cite|improve this answer





















  • Got minimum value 1/4. But Maximum ?? By plotting graph its easy, but is there any mathematical method ??
    – Alvin Carter
    Nov 20 at 5:35










  • @AlvinCarter Have you been taught derivatives?
    – Jimmy R.
    Nov 20 at 5:58












  • Ya. I did it with both by AM-GM inequality as well as derivative equating to zero. But, it giving me the maximum value. How can I find minimum ?
    – Alvin Carter
    Nov 20 at 6:13










  • Did you check the value of the function at the edges of the acceptable domain? Does this ring a bell?
    – Jimmy R.
    Nov 20 at 6:45






  • 1




    Oh... Now I got it. At one of the edge , the Minimum is -2. Thanks ! Cheers.
    – Alvin Carter
    Nov 20 at 8:21


















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










Hint: $|a|ge 1$ implies that $-1le ale 1$. Also, $a+b=1$ implies that $b=1-a$, hence $$ab=a(1-a)$$



So, your task is to find the minimum and the maximum of the function $f(a)=a(1-a)$ in the interval $ain [-1,1]$. Can you do it?






share|cite|improve this answer





















  • Got minimum value 1/4. But Maximum ?? By plotting graph its easy, but is there any mathematical method ??
    – Alvin Carter
    Nov 20 at 5:35










  • @AlvinCarter Have you been taught derivatives?
    – Jimmy R.
    Nov 20 at 5:58












  • Ya. I did it with both by AM-GM inequality as well as derivative equating to zero. But, it giving me the maximum value. How can I find minimum ?
    – Alvin Carter
    Nov 20 at 6:13










  • Did you check the value of the function at the edges of the acceptable domain? Does this ring a bell?
    – Jimmy R.
    Nov 20 at 6:45






  • 1




    Oh... Now I got it. At one of the edge , the Minimum is -2. Thanks ! Cheers.
    – Alvin Carter
    Nov 20 at 8:21















up vote
1
down vote



accepted










Hint: $|a|ge 1$ implies that $-1le ale 1$. Also, $a+b=1$ implies that $b=1-a$, hence $$ab=a(1-a)$$



So, your task is to find the minimum and the maximum of the function $f(a)=a(1-a)$ in the interval $ain [-1,1]$. Can you do it?






share|cite|improve this answer





















  • Got minimum value 1/4. But Maximum ?? By plotting graph its easy, but is there any mathematical method ??
    – Alvin Carter
    Nov 20 at 5:35










  • @AlvinCarter Have you been taught derivatives?
    – Jimmy R.
    Nov 20 at 5:58












  • Ya. I did it with both by AM-GM inequality as well as derivative equating to zero. But, it giving me the maximum value. How can I find minimum ?
    – Alvin Carter
    Nov 20 at 6:13










  • Did you check the value of the function at the edges of the acceptable domain? Does this ring a bell?
    – Jimmy R.
    Nov 20 at 6:45






  • 1




    Oh... Now I got it. At one of the edge , the Minimum is -2. Thanks ! Cheers.
    – Alvin Carter
    Nov 20 at 8:21













up vote
1
down vote



accepted







up vote
1
down vote



accepted






Hint: $|a|ge 1$ implies that $-1le ale 1$. Also, $a+b=1$ implies that $b=1-a$, hence $$ab=a(1-a)$$



So, your task is to find the minimum and the maximum of the function $f(a)=a(1-a)$ in the interval $ain [-1,1]$. Can you do it?






share|cite|improve this answer












Hint: $|a|ge 1$ implies that $-1le ale 1$. Also, $a+b=1$ implies that $b=1-a$, hence $$ab=a(1-a)$$



So, your task is to find the minimum and the maximum of the function $f(a)=a(1-a)$ in the interval $ain [-1,1]$. Can you do it?







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 20 at 4:28









Jimmy R.

33k42157




33k42157












  • Got minimum value 1/4. But Maximum ?? By plotting graph its easy, but is there any mathematical method ??
    – Alvin Carter
    Nov 20 at 5:35










  • @AlvinCarter Have you been taught derivatives?
    – Jimmy R.
    Nov 20 at 5:58












  • Ya. I did it with both by AM-GM inequality as well as derivative equating to zero. But, it giving me the maximum value. How can I find minimum ?
    – Alvin Carter
    Nov 20 at 6:13










  • Did you check the value of the function at the edges of the acceptable domain? Does this ring a bell?
    – Jimmy R.
    Nov 20 at 6:45






  • 1




    Oh... Now I got it. At one of the edge , the Minimum is -2. Thanks ! Cheers.
    – Alvin Carter
    Nov 20 at 8:21


















  • Got minimum value 1/4. But Maximum ?? By plotting graph its easy, but is there any mathematical method ??
    – Alvin Carter
    Nov 20 at 5:35










  • @AlvinCarter Have you been taught derivatives?
    – Jimmy R.
    Nov 20 at 5:58












  • Ya. I did it with both by AM-GM inequality as well as derivative equating to zero. But, it giving me the maximum value. How can I find minimum ?
    – Alvin Carter
    Nov 20 at 6:13










  • Did you check the value of the function at the edges of the acceptable domain? Does this ring a bell?
    – Jimmy R.
    Nov 20 at 6:45






  • 1




    Oh... Now I got it. At one of the edge , the Minimum is -2. Thanks ! Cheers.
    – Alvin Carter
    Nov 20 at 8:21
















Got minimum value 1/4. But Maximum ?? By plotting graph its easy, but is there any mathematical method ??
– Alvin Carter
Nov 20 at 5:35




Got minimum value 1/4. But Maximum ?? By plotting graph its easy, but is there any mathematical method ??
– Alvin Carter
Nov 20 at 5:35












@AlvinCarter Have you been taught derivatives?
– Jimmy R.
Nov 20 at 5:58






@AlvinCarter Have you been taught derivatives?
– Jimmy R.
Nov 20 at 5:58














Ya. I did it with both by AM-GM inequality as well as derivative equating to zero. But, it giving me the maximum value. How can I find minimum ?
– Alvin Carter
Nov 20 at 6:13




Ya. I did it with both by AM-GM inequality as well as derivative equating to zero. But, it giving me the maximum value. How can I find minimum ?
– Alvin Carter
Nov 20 at 6:13












Did you check the value of the function at the edges of the acceptable domain? Does this ring a bell?
– Jimmy R.
Nov 20 at 6:45




Did you check the value of the function at the edges of the acceptable domain? Does this ring a bell?
– Jimmy R.
Nov 20 at 6:45




1




1




Oh... Now I got it. At one of the edge , the Minimum is -2. Thanks ! Cheers.
– Alvin Carter
Nov 20 at 8:21




Oh... Now I got it. At one of the edge , the Minimum is -2. Thanks ! Cheers.
– Alvin Carter
Nov 20 at 8:21



-

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