Min value of a trigonometric expression

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












What is the value of $sin(x)$ for the maximum value of $(5+3sin(x))^2 (7-3sin(x))^3$.






calculus optimization







share|cite|improve this question



























    up vote
    1
    down vote

    favorite












    What is the value of $sin(x)$ for the maximum value of $(5+3sin(x))^2 (7-3sin(x))^3$.






    calculus optimization







    share|cite|improve this question

























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      What is the value of $sin(x)$ for the maximum value of $(5+3sin(x))^2 (7-3sin(x))^3$.






      calculus optimization







      share|cite|improve this question















      What is the value of $sin(x)$ for the maximum value of $(5+3sin(x))^2 (7-3sin(x))^3$.






      calculus optimization




      calculus optimization






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 1 hour ago









      gt6989b

      31.5k22349




      31.5k22349










      asked 1 hour ago









      Hik Aubergine

      185




      185




















          3 Answers
          3






          active

          oldest

          votes

















          up vote
          4
          down vote



          accepted










          Hint:



          AM GM inequality



          $$dfrac2cdot3(5+3sin x)+3cdot2(7-3sin x)2+3gesqrt[5]3^2(5+3sin x)^22^3(7-3sin x)^3$$



          The equality occurs if $3(5+3sin x)=2(7-3sin x)$






          share|cite|improve this answer




















          • Thanks a lot sir, thats what im looking for
            – Hik Aubergine
            1 hour ago

















          up vote
          1
          down vote













          AM-GM inequality is the neat approach, but if you can't figure it out algebra also helps...



          let $z=sin(x)$, $f(z)=g(z)^2h(z)^3$



          $$
          f' = 2gg'h^3 + 3gh^2h' = gh^2(2g'h+3hh')
          $$



          since $-1 le z le 1$, $g$ and $h$ do not have zeros in this region. Therefore $2g'h=-3hh'$, which will give you the same equation to solve for $z$.






          share|cite|improve this answer



























            up vote
            0
            down vote













            HINT



            Let $z = sin x$. You are asking for what value of $z$ does the function $$f(z) = (5+3z)^2 (7-3z)^3$$ attains its maximum. Taking a derivative would likely help.



            UPDATE



            As mentioned in the comments below, you are only optimizing over $-1 le z le 1$.






            share|cite|improve this answer


















            • 1




              Worth mentioning that $x$ is probably meant to be real, so $z$ has to be between $-1$ and $1$.
              – SmileyCraft
              1 hour ago










            Your Answer




            StackExchange.ifUsing("editor", function ()
            return StackExchange.using("mathjaxEditing", function ()
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            );
            );
            , "mathjax-editing");

            StackExchange.ready(function()
            var channelOptions =
            tags: "".split(" "),
            id: "69"
            ;
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function()
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled)
            StackExchange.using("snippets", function()
            createEditor();
            );

            else
            createEditor();

            );

            function createEditor()
            StackExchange.prepareEditor(
            heartbeatType: 'answer',
            convertImagesToLinks: true,
            noModals: false,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            );



            );













             

            draft saved


            draft discarded


















            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2958298%2fmin-value-of-a-trigonometric-expression%23new-answer', 'question_page');

            );

            Post as a guest

















            3 Answers
            3






            active

            oldest

            votes








            3 Answers
            3






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            4
            down vote



            accepted










            Hint:



            AM GM inequality



            $$dfrac2cdot3(5+3sin x)+3cdot2(7-3sin x)2+3gesqrt[5]3^2(5+3sin x)^22^3(7-3sin x)^3$$



            The equality occurs if $3(5+3sin x)=2(7-3sin x)$






            share|cite|improve this answer




















            • Thanks a lot sir, thats what im looking for
              – Hik Aubergine
              1 hour ago














            up vote
            4
            down vote



            accepted










            Hint:



            AM GM inequality



            $$dfrac2cdot3(5+3sin x)+3cdot2(7-3sin x)2+3gesqrt[5]3^2(5+3sin x)^22^3(7-3sin x)^3$$



            The equality occurs if $3(5+3sin x)=2(7-3sin x)$






            share|cite|improve this answer




















            • Thanks a lot sir, thats what im looking for
              – Hik Aubergine
              1 hour ago












            up vote
            4
            down vote



            accepted







            up vote
            4
            down vote



            accepted






            Hint:



            AM GM inequality



            $$dfrac2cdot3(5+3sin x)+3cdot2(7-3sin x)2+3gesqrt[5]3^2(5+3sin x)^22^3(7-3sin x)^3$$



            The equality occurs if $3(5+3sin x)=2(7-3sin x)$






            share|cite|improve this answer












            Hint:



            AM GM inequality



            $$dfrac2cdot3(5+3sin x)+3cdot2(7-3sin x)2+3gesqrt[5]3^2(5+3sin x)^22^3(7-3sin x)^3$$



            The equality occurs if $3(5+3sin x)=2(7-3sin x)$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 1 hour ago









            lab bhattacharjee

            217k14153267




            217k14153267











            • Thanks a lot sir, thats what im looking for
              – Hik Aubergine
              1 hour ago
















            • Thanks a lot sir, thats what im looking for
              – Hik Aubergine
              1 hour ago















            Thanks a lot sir, thats what im looking for
            – Hik Aubergine
            1 hour ago




            Thanks a lot sir, thats what im looking for
            – Hik Aubergine
            1 hour ago










            up vote
            1
            down vote













            AM-GM inequality is the neat approach, but if you can't figure it out algebra also helps...



            let $z=sin(x)$, $f(z)=g(z)^2h(z)^3$



            $$
            f' = 2gg'h^3 + 3gh^2h' = gh^2(2g'h+3hh')
            $$



            since $-1 le z le 1$, $g$ and $h$ do not have zeros in this region. Therefore $2g'h=-3hh'$, which will give you the same equation to solve for $z$.






            share|cite|improve this answer
























              up vote
              1
              down vote













              AM-GM inequality is the neat approach, but if you can't figure it out algebra also helps...



              let $z=sin(x)$, $f(z)=g(z)^2h(z)^3$



              $$
              f' = 2gg'h^3 + 3gh^2h' = gh^2(2g'h+3hh')
              $$



              since $-1 le z le 1$, $g$ and $h$ do not have zeros in this region. Therefore $2g'h=-3hh'$, which will give you the same equation to solve for $z$.






              share|cite|improve this answer






















                up vote
                1
                down vote










                up vote
                1
                down vote









                AM-GM inequality is the neat approach, but if you can't figure it out algebra also helps...



                let $z=sin(x)$, $f(z)=g(z)^2h(z)^3$



                $$
                f' = 2gg'h^3 + 3gh^2h' = gh^2(2g'h+3hh')
                $$



                since $-1 le z le 1$, $g$ and $h$ do not have zeros in this region. Therefore $2g'h=-3hh'$, which will give you the same equation to solve for $z$.






                share|cite|improve this answer












                AM-GM inequality is the neat approach, but if you can't figure it out algebra also helps...



                let $z=sin(x)$, $f(z)=g(z)^2h(z)^3$



                $$
                f' = 2gg'h^3 + 3gh^2h' = gh^2(2g'h+3hh')
                $$



                since $-1 le z le 1$, $g$ and $h$ do not have zeros in this region. Therefore $2g'h=-3hh'$, which will give you the same equation to solve for $z$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 1 hour ago









                karakfa

                1,813811




                1,813811




















                    up vote
                    0
                    down vote













                    HINT



                    Let $z = sin x$. You are asking for what value of $z$ does the function $$f(z) = (5+3z)^2 (7-3z)^3$$ attains its maximum. Taking a derivative would likely help.



                    UPDATE



                    As mentioned in the comments below, you are only optimizing over $-1 le z le 1$.






                    share|cite|improve this answer


















                    • 1




                      Worth mentioning that $x$ is probably meant to be real, so $z$ has to be between $-1$ and $1$.
                      – SmileyCraft
                      1 hour ago














                    up vote
                    0
                    down vote













                    HINT



                    Let $z = sin x$. You are asking for what value of $z$ does the function $$f(z) = (5+3z)^2 (7-3z)^3$$ attains its maximum. Taking a derivative would likely help.



                    UPDATE



                    As mentioned in the comments below, you are only optimizing over $-1 le z le 1$.






                    share|cite|improve this answer


















                    • 1




                      Worth mentioning that $x$ is probably meant to be real, so $z$ has to be between $-1$ and $1$.
                      – SmileyCraft
                      1 hour ago












                    up vote
                    0
                    down vote










                    up vote
                    0
                    down vote









                    HINT



                    Let $z = sin x$. You are asking for what value of $z$ does the function $$f(z) = (5+3z)^2 (7-3z)^3$$ attains its maximum. Taking a derivative would likely help.



                    UPDATE



                    As mentioned in the comments below, you are only optimizing over $-1 le z le 1$.






                    share|cite|improve this answer














                    HINT



                    Let $z = sin x$. You are asking for what value of $z$ does the function $$f(z) = (5+3z)^2 (7-3z)^3$$ attains its maximum. Taking a derivative would likely help.



                    UPDATE



                    As mentioned in the comments below, you are only optimizing over $-1 le z le 1$.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited 44 mins ago

























                    answered 1 hour ago









                    gt6989b

                    31.5k22349




                    31.5k22349







                    • 1




                      Worth mentioning that $x$ is probably meant to be real, so $z$ has to be between $-1$ and $1$.
                      – SmileyCraft
                      1 hour ago












                    • 1




                      Worth mentioning that $x$ is probably meant to be real, so $z$ has to be between $-1$ and $1$.
                      – SmileyCraft
                      1 hour ago







                    1




                    1




                    Worth mentioning that $x$ is probably meant to be real, so $z$ has to be between $-1$ and $1$.
                    – SmileyCraft
                    1 hour ago




                    Worth mentioning that $x$ is probably meant to be real, so $z$ has to be between $-1$ and $1$.
                    – SmileyCraft
                    1 hour ago

















                     

                    draft saved


                    draft discarded















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2958298%2fmin-value-of-a-trigonometric-expression%23new-answer', 'question_page');

                    );

                    Post as a guest
































































                    -

                    Popular posts from this blog

                    How to check contact read email or not when send email to Individual?

                    Image
                    Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite I'm using WordPress 4.9.8, CiviCRM to 5.5.1, I usually send email to contact by Search> Find contacts View contact details Action> Send email Send email ok, Contact received mail ok like picture But status only Email sent though contact read email or not. So, can CiviCRM can change status to Email read when contact read email? wordpress email share | improve this question asked Sep 26 at 0:12 ToanLuong 49 9 add a comment  |  up vote 1 down vote favorite I'm using WordPress 4.9.8, CiviCRM to 5.5.1, I usually send email to contact by Search> Find contacts View contact details Action> Send email Send email ok, Contact received mail ok like picture But status only Email sent though contact read email or not. So, can CiviCRM can change status to Email read when contact read email? wordpress email share | improve this question asked Se
                    Read more

                    Displaying single band from multi-band raster using QGIS

                    Image
                    Clash Royale CLAN TAG #URR8PPP 1 How can I extract a single band from multi-band raster in QGIS? I have an remote sensed image which has 6 bands (including NDVI band), I want to display each band separately, but have no idea how to do. I have seen some questions similar here but none worked for me. The original image (has 6 bands) is: I want to display the band 6 which should be like this: But I tried gdal_translate, and couldn't get the correct result. What I have got is: qgis raster multi-band share | improve this question edited Mar 5 at 0:53 Summer asked Mar 4 at 6:42 Summer Summer 23 6 Is this any help gis.stackexchange.com/questions/220658/… ? if not gis.stackexchange.com/questions/62133/… might help. – Michael Stimson Mar 4 at 6:46 Thanks for answering but when I used gdal_translate, qgis showed that 'Error 4: Kayena.tif: No such file or directory". Would you know how to fix it? –
                    Read more

                    How many registers does an x86_64 CPU actually have?

                    Image
                    Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite I am currently learning reverse engineering and am studying the flags register. I had in my mind that rflags was just another name for one of the 16 general purpose registers, for example rax or rbx . But it looks like rflags is actually an additional register. So that makes 17 registers in total... how many more could there be? I have spent at least an hour on this and found numerous different answers. The best answer so far is this, which says that there are 40 registers in total. 16 General Purpose Registers 2 Status Registers 6 Code Segment Registers 16 SSE Registers 8 FPU/MMX Registers But if I add that up, I get 48. Could anybody provide an official answer on how many registers an x86_64 CPU has (e.g. an Intel i7). Additionally, I have seen references to 'hardware' and 'architectural' registers. What are those registers and how many are there? register x86-64 share | improve this
                    Read more