Min value of a trigonometric expression

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






calculus optimization







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    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 2 hours ago









      gt6989b

      31.5k22349




      31.5k22349










      asked 2 hours 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
              2 hours ago










            Your Answer




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            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 2 hours 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
                      2 hours 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
                      2 hours 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 55 mins ago

























                    answered 2 hours 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
                      2 hours ago












                    • 1




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







                    1




                    1




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




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

















                     

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