$P(x)$ is a polynomial of least degree with a local maximum of $6$ at $x=1$ and minimum of $2$ at $x=3$. Find $P^prime(0)$.

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











up vote
2
down vote

favorite













Let $P(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and minimum at $x=3$.If $P(1)=6$ and $P(3)=2$, then find $P'(0)$.




Method: Since the smallest polynomial whose derivative gives 2 root would be a cubic equation. So I assumed my function to be



$$P(x) = ax^3+ bx^2+cx+d$$



Solving the condition above gives me $b=-6a$ and $c=11a$. I am stuck after this. Can anyone tell me how to proceed from here?






calculus







share|cite|improve this question



















  • 2




    Well, you also know that $P(1)=6,P(3)=2$.
    – lulu
    25 mins ago






  • 1




    Note: I don't agree that $c=11a$. You have $P'(x)=3ax^2+2bx+c=lambda(x-3)(x-1)=lambda x^2-4lambda x+3lambdaimplies a=3lambda,,2b=-4lambda=-12a,,c=3lambda=9a$. No?
    – lulu
    21 mins ago















up vote
2
down vote

favorite













Let $P(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and minimum at $x=3$.If $P(1)=6$ and $P(3)=2$, then find $P'(0)$.




Method: Since the smallest polynomial whose derivative gives 2 root would be a cubic equation. So I assumed my function to be



$$P(x) = ax^3+ bx^2+cx+d$$



Solving the condition above gives me $b=-6a$ and $c=11a$. I am stuck after this. Can anyone tell me how to proceed from here?






calculus







share|cite|improve this question



















  • 2




    Well, you also know that $P(1)=6,P(3)=2$.
    – lulu
    25 mins ago






  • 1




    Note: I don't agree that $c=11a$. You have $P'(x)=3ax^2+2bx+c=lambda(x-3)(x-1)=lambda x^2-4lambda x+3lambdaimplies a=3lambda,,2b=-4lambda=-12a,,c=3lambda=9a$. No?
    – lulu
    21 mins ago













up vote
2
down vote

favorite









up vote
2
down vote

favorite












Let $P(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and minimum at $x=3$.If $P(1)=6$ and $P(3)=2$, then find $P'(0)$.




Method: Since the smallest polynomial whose derivative gives 2 root would be a cubic equation. So I assumed my function to be



$$P(x) = ax^3+ bx^2+cx+d$$



Solving the condition above gives me $b=-6a$ and $c=11a$. I am stuck after this. Can anyone tell me how to proceed from here?






calculus







share|cite|improve this question
















Let $P(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and minimum at $x=3$.If $P(1)=6$ and $P(3)=2$, then find $P'(0)$.




Method: Since the smallest polynomial whose derivative gives 2 root would be a cubic equation. So I assumed my function to be



$$P(x) = ax^3+ bx^2+cx+d$$



Solving the condition above gives me $b=-6a$ and $c=11a$. I am stuck after this. Can anyone tell me how to proceed from here?






calculus




calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 7 mins ago









Blue

44.4k868142




44.4k868142










asked 26 mins ago









Abhinav

1325




1325







  • 2




    Well, you also know that $P(1)=6,P(3)=2$.
    – lulu
    25 mins ago






  • 1




    Note: I don't agree that $c=11a$. You have $P'(x)=3ax^2+2bx+c=lambda(x-3)(x-1)=lambda x^2-4lambda x+3lambdaimplies a=3lambda,,2b=-4lambda=-12a,,c=3lambda=9a$. No?
    – lulu
    21 mins ago













  • 2




    Well, you also know that $P(1)=6,P(3)=2$.
    – lulu
    25 mins ago






  • 1




    Note: I don't agree that $c=11a$. You have $P'(x)=3ax^2+2bx+c=lambda(x-3)(x-1)=lambda x^2-4lambda x+3lambdaimplies a=3lambda,,2b=-4lambda=-12a,,c=3lambda=9a$. No?
    – lulu
    21 mins ago








2




2




Well, you also know that $P(1)=6,P(3)=2$.
– lulu
25 mins ago




Well, you also know that $P(1)=6,P(3)=2$.
– lulu
25 mins ago




1




1




Note: I don't agree that $c=11a$. You have $P'(x)=3ax^2+2bx+c=lambda(x-3)(x-1)=lambda x^2-4lambda x+3lambdaimplies a=3lambda,,2b=-4lambda=-12a,,c=3lambda=9a$. No?
– lulu
21 mins ago





Note: I don't agree that $c=11a$. You have $P'(x)=3ax^2+2bx+c=lambda(x-3)(x-1)=lambda x^2-4lambda x+3lambdaimplies a=3lambda,,2b=-4lambda=-12a,,c=3lambda=9a$. No?
– lulu
21 mins ago











2 Answers
2






active

oldest

votes

















up vote
3
down vote



accepted










Hint: Consider the System
$$a+b+c+d=6$$
$$27a+9b+3c+d=2$$
$$3a+2b+c=0$$
$$27a+6b+c=0$$






share|cite|improve this answer



























    up vote
    2
    down vote













    HINT



    You attempt was right to assume a function of the type $P(x)=ax^3+bx^2+cx+d$ as the solution. Rewriting the conditions as $P(1)=6$,$P(3)=2$,$P'(1)=0$ and $P'(3)=0$ leads to a system of four equations with four variables $a,b,c$ and $d$.



    Can you do it on your own from hereon?






    share|cite|improve this answer




















      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%2f2927691%2fpx-is-a-polynomial-of-least-degree-with-a-local-maximum-of-6-at-x-1-and%23new-answer', 'question_page');

      );

      Post as a guest

















      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      3
      down vote



      accepted










      Hint: Consider the System
      $$a+b+c+d=6$$
      $$27a+9b+3c+d=2$$
      $$3a+2b+c=0$$
      $$27a+6b+c=0$$






      share|cite|improve this answer
























        up vote
        3
        down vote



        accepted










        Hint: Consider the System
        $$a+b+c+d=6$$
        $$27a+9b+3c+d=2$$
        $$3a+2b+c=0$$
        $$27a+6b+c=0$$






        share|cite|improve this answer






















          up vote
          3
          down vote



          accepted







          up vote
          3
          down vote



          accepted






          Hint: Consider the System
          $$a+b+c+d=6$$
          $$27a+9b+3c+d=2$$
          $$3a+2b+c=0$$
          $$27a+6b+c=0$$






          share|cite|improve this answer












          Hint: Consider the System
          $$a+b+c+d=6$$
          $$27a+9b+3c+d=2$$
          $$3a+2b+c=0$$
          $$27a+6b+c=0$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 21 mins ago









          Dr. Sonnhard Graubner

          69.2k32761




          69.2k32761




















              up vote
              2
              down vote













              HINT



              You attempt was right to assume a function of the type $P(x)=ax^3+bx^2+cx+d$ as the solution. Rewriting the conditions as $P(1)=6$,$P(3)=2$,$P'(1)=0$ and $P'(3)=0$ leads to a system of four equations with four variables $a,b,c$ and $d$.



              Can you do it on your own from hereon?






              share|cite|improve this answer
























                up vote
                2
                down vote













                HINT



                You attempt was right to assume a function of the type $P(x)=ax^3+bx^2+cx+d$ as the solution. Rewriting the conditions as $P(1)=6$,$P(3)=2$,$P'(1)=0$ and $P'(3)=0$ leads to a system of four equations with four variables $a,b,c$ and $d$.



                Can you do it on your own from hereon?






                share|cite|improve this answer






















                  up vote
                  2
                  down vote










                  up vote
                  2
                  down vote









                  HINT



                  You attempt was right to assume a function of the type $P(x)=ax^3+bx^2+cx+d$ as the solution. Rewriting the conditions as $P(1)=6$,$P(3)=2$,$P'(1)=0$ and $P'(3)=0$ leads to a system of four equations with four variables $a,b,c$ and $d$.



                  Can you do it on your own from hereon?






                  share|cite|improve this answer












                  HINT



                  You attempt was right to assume a function of the type $P(x)=ax^3+bx^2+cx+d$ as the solution. Rewriting the conditions as $P(1)=6$,$P(3)=2$,$P'(1)=0$ and $P'(3)=0$ leads to a system of four equations with four variables $a,b,c$ and $d$.



                  Can you do it on your own from hereon?







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 20 mins ago









                  mrtaurho

                  801219




                  801219



























                       

                      draft saved


                      draft discarded















































                       


                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function ()
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2927691%2fpx-is-a-polynomial-of-least-degree-with-a-local-maximum-of-6-at-x-1-and%23new-answer', 'question_page');

                      );

                      Post as a guest
































































                      Comments

                      Post a Comment

                      Popular posts from this blog

                      Long meetings (6-7 hours a day): Being “babysat” by supervisor

                      Image
                      Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote 31 down vote favorite 4 I hold a PhD in mechanical engineering with a 7 year background in self-guided, self-paced research including 2 years as a graduate level course instructor and a subject matter expert for various engineering projects in a university setting in the United States. For the last 10-11 months, I am working in a research industry in France where my task is to debug C++ spaghetti code written over a period of several years by my current supervisor. It is a team of 2: just I and my supervisor. Diurnally, I am posed with a "bug of the day" to get out of this C++ code. My supervisor generally gives me about 15-30 minutes to solve it and then accosts me about whether or not the task is completed and then rinses and repeats this until the end of the day. Off-late, my supervisor has been holding 6-7 hour long "meetin
                      Read more

                      Is the Concept of Multiple Fantasy Races Scientifically Flawed? [closed]

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite Many fantasy worlds have multiple humanoid species (elves, orcs, humans, etc) living in the same world, in addition to that, they often have the ability to interbreed. And I can't help but question their believability... fantasy-races reproduction interspecies-relations share | improve this question edited Aug 9 at 14:46 Michael Kjörling ♦ 21.1k 10 85 161 asked Aug 9 at 13:13 Chlodio 118 6 closed as unclear what you're asking by MichaelK, Gryphon, John, Vincent, Frostfyre Aug 9 at 15:25 Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. 3 Stack Exchange exist to
                      Read more

                      Confectionery

                      Image
                      Prepared foods made with a lot of sugar or other cabohydrates "Sweetmeat" redirects here. For the racehorse, see Sweetmeat (horse). Not to be confused with Sweetbread. This Kransekake is a traditional Scandinavian baker's confection. Confectionery is the art of making confections , which are food items that are rich in sugar and carbohydrates. Exact definitions are difficult. [1] In general, though, confectionery is divided into two broad and somewhat overlapping categories, bakers' confections and sugar confections. [2] Bakers' confectionery, also called flour confections , includes principally sweet pastries, cakes, and similar baked goods. Sugar confectionery includes candies ( sweets in British English), candied nuts, chocolates, chewing gum, bubble gum, pastillage, and other confections that are made primarily of sugar. In some cases, chocolate confections (confections made of chocolate) are treated as a separate category, as are sugar-free versions of
                      Read more