Find the number of roots of $F(x)= int_0^x e^t(t^2-3t-5)mathrm dt , x>0$ in the interval $(0,4)$












3












$begingroup$



Let $$F(x)= int_0^x e^t(t^2-3t-5)mathrm dt , x>0$$
Find the number of roots of $F(x)=0$ in the interval $(0,4)$.




My attempt: I simply integrated it and got $F(x)=e^x(x(x-5))$ which has roots $0$ and $5$ and none is between $0$ and $4$. So the answer is $0$, which is correct.



My question: Is there any other way to do this question in comparatively less time?






integration roots







share|cite|improve this question











$endgroup$

















    3












    $begingroup$



    Let $$F(x)= int_0^x e^t(t^2-3t-5)mathrm dt , x>0$$
    Find the number of roots of $F(x)=0$ in the interval $(0,4)$.




    My attempt: I simply integrated it and got $F(x)=e^x(x(x-5))$ which has roots $0$ and $5$ and none is between $0$ and $4$. So the answer is $0$, which is correct.



    My question: Is there any other way to do this question in comparatively less time?






    integration roots







    share|cite|improve this question











    $endgroup$















      3












      3








      3





      $begingroup$



      Let $$F(x)= int_0^x e^t(t^2-3t-5)mathrm dt , x>0$$
      Find the number of roots of $F(x)=0$ in the interval $(0,4)$.




      My attempt: I simply integrated it and got $F(x)=e^x(x(x-5))$ which has roots $0$ and $5$ and none is between $0$ and $4$. So the answer is $0$, which is correct.



      My question: Is there any other way to do this question in comparatively less time?






      integration roots







      share|cite|improve this question











      $endgroup$





      Let $$F(x)= int_0^x e^t(t^2-3t-5)mathrm dt , x>0$$
      Find the number of roots of $F(x)=0$ in the interval $(0,4)$.




      My attempt: I simply integrated it and got $F(x)=e^x(x(x-5))$ which has roots $0$ and $5$ and none is between $0$ and $4$. So the answer is $0$, which is correct.



      My question: Is there any other way to do this question in comparatively less time?






      integration roots




      integration roots






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 15 at 10:51









      mrtaurho

      6,19571641




      6,19571641










      asked Jan 15 at 10:48









      user233797user233797

      162




      162






















          3 Answers
          3






          active

          oldest

          votes


















          4












          $begingroup$

          You have $F(0)=0, F'(x)=e^x(x^2-3x-5)<0$ in $(0,4)$. This means $F(x)$ is strictly decreasing in $(0,4)$, so that $F(x)<0$ in $(0,4)$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you. It was really helpful.
            $endgroup$
            – user233797
            Jan 15 at 11:01










          • $begingroup$
            You are welcome
            $endgroup$
            – Shubham Johri
            Jan 15 at 11:02



















          2












          $begingroup$

          Hint: note that
          $$x^2-3x-5<0$$
          On the interval $(0, 4)$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Okay I got it. Thank you.
            $endgroup$
            – user233797
            Jan 15 at 11:02



















          2












          $begingroup$

          Yes: it's easy to see, without computing the roots, that $t^2-3t-5$ has a negative and a positive root, and the latter is $>4$. Using the First Fundamental Theorem of Integral Calculus, $F(x)$ decreases on the interval $(0,4)$. Now $F(0)=0$, hence $F(x)<0$ for $0<x<4$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you. I got it now.
            $endgroup$
            – user233797
            Jan 15 at 11:24












          Your Answer








          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',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          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%2f3074308%2ffind-the-number-of-roots-of-fx-int-0x-ett2-3t-5-mathrm-dt-x0-in-th%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          3 Answers
          3






          active

          oldest

          votes








          3 Answers
          3






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          4












          $begingroup$

          You have $F(0)=0, F'(x)=e^x(x^2-3x-5)<0$ in $(0,4)$. This means $F(x)$ is strictly decreasing in $(0,4)$, so that $F(x)<0$ in $(0,4)$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you. It was really helpful.
            $endgroup$
            – user233797
            Jan 15 at 11:01










          • $begingroup$
            You are welcome
            $endgroup$
            – Shubham Johri
            Jan 15 at 11:02
















          4












          $begingroup$

          You have $F(0)=0, F'(x)=e^x(x^2-3x-5)<0$ in $(0,4)$. This means $F(x)$ is strictly decreasing in $(0,4)$, so that $F(x)<0$ in $(0,4)$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you. It was really helpful.
            $endgroup$
            – user233797
            Jan 15 at 11:01










          • $begingroup$
            You are welcome
            $endgroup$
            – Shubham Johri
            Jan 15 at 11:02














          4












          4








          4





          $begingroup$

          You have $F(0)=0, F'(x)=e^x(x^2-3x-5)<0$ in $(0,4)$. This means $F(x)$ is strictly decreasing in $(0,4)$, so that $F(x)<0$ in $(0,4)$.






          share|cite|improve this answer









          $endgroup$



          You have $F(0)=0, F'(x)=e^x(x^2-3x-5)<0$ in $(0,4)$. This means $F(x)$ is strictly decreasing in $(0,4)$, so that $F(x)<0$ in $(0,4)$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 15 at 10:57









          Shubham JohriShubham Johri

          5,668918




          5,668918












          • $begingroup$
            Thank you. It was really helpful.
            $endgroup$
            – user233797
            Jan 15 at 11:01










          • $begingroup$
            You are welcome
            $endgroup$
            – Shubham Johri
            Jan 15 at 11:02


















          • $begingroup$
            Thank you. It was really helpful.
            $endgroup$
            – user233797
            Jan 15 at 11:01










          • $begingroup$
            You are welcome
            $endgroup$
            – Shubham Johri
            Jan 15 at 11:02
















          $begingroup$
          Thank you. It was really helpful.
          $endgroup$
          – user233797
          Jan 15 at 11:01




          $begingroup$
          Thank you. It was really helpful.
          $endgroup$
          – user233797
          Jan 15 at 11:01












          $begingroup$
          You are welcome
          $endgroup$
          – Shubham Johri
          Jan 15 at 11:02




          $begingroup$
          You are welcome
          $endgroup$
          – Shubham Johri
          Jan 15 at 11:02











          2












          $begingroup$

          Hint: note that
          $$x^2-3x-5<0$$
          On the interval $(0, 4)$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Okay I got it. Thank you.
            $endgroup$
            – user233797
            Jan 15 at 11:02
















          2












          $begingroup$

          Hint: note that
          $$x^2-3x-5<0$$
          On the interval $(0, 4)$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Okay I got it. Thank you.
            $endgroup$
            – user233797
            Jan 15 at 11:02














          2












          2








          2





          $begingroup$

          Hint: note that
          $$x^2-3x-5<0$$
          On the interval $(0, 4)$






          share|cite|improve this answer









          $endgroup$



          Hint: note that
          $$x^2-3x-5<0$$
          On the interval $(0, 4)$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 15 at 10:57









          BotondBotond

          6,69531034




          6,69531034












          • $begingroup$
            Okay I got it. Thank you.
            $endgroup$
            – user233797
            Jan 15 at 11:02


















          • $begingroup$
            Okay I got it. Thank you.
            $endgroup$
            – user233797
            Jan 15 at 11:02
















          $begingroup$
          Okay I got it. Thank you.
          $endgroup$
          – user233797
          Jan 15 at 11:02




          $begingroup$
          Okay I got it. Thank you.
          $endgroup$
          – user233797
          Jan 15 at 11:02











          2












          $begingroup$

          Yes: it's easy to see, without computing the roots, that $t^2-3t-5$ has a negative and a positive root, and the latter is $>4$. Using the First Fundamental Theorem of Integral Calculus, $F(x)$ decreases on the interval $(0,4)$. Now $F(0)=0$, hence $F(x)<0$ for $0<x<4$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you. I got it now.
            $endgroup$
            – user233797
            Jan 15 at 11:24
















          2












          $begingroup$

          Yes: it's easy to see, without computing the roots, that $t^2-3t-5$ has a negative and a positive root, and the latter is $>4$. Using the First Fundamental Theorem of Integral Calculus, $F(x)$ decreases on the interval $(0,4)$. Now $F(0)=0$, hence $F(x)<0$ for $0<x<4$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you. I got it now.
            $endgroup$
            – user233797
            Jan 15 at 11:24














          2












          2








          2





          $begingroup$

          Yes: it's easy to see, without computing the roots, that $t^2-3t-5$ has a negative and a positive root, and the latter is $>4$. Using the First Fundamental Theorem of Integral Calculus, $F(x)$ decreases on the interval $(0,4)$. Now $F(0)=0$, hence $F(x)<0$ for $0<x<4$.






          share|cite|improve this answer









          $endgroup$



          Yes: it's easy to see, without computing the roots, that $t^2-3t-5$ has a negative and a positive root, and the latter is $>4$. Using the First Fundamental Theorem of Integral Calculus, $F(x)$ decreases on the interval $(0,4)$. Now $F(0)=0$, hence $F(x)<0$ for $0<x<4$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 15 at 11:04









          BernardBernard

          124k743118




          124k743118












          • $begingroup$
            Thank you. I got it now.
            $endgroup$
            – user233797
            Jan 15 at 11:24


















          • $begingroup$
            Thank you. I got it now.
            $endgroup$
            – user233797
            Jan 15 at 11:24
















          $begingroup$
          Thank you. I got it now.
          $endgroup$
          – user233797
          Jan 15 at 11:24




          $begingroup$
          Thank you. I got it now.
          $endgroup$
          – user233797
          Jan 15 at 11:24


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3074308%2ffind-the-number-of-roots-of-fx-int-0x-ett2-3t-5-mathrm-dt-x0-in-th%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          -

          Popular posts from this blog

          Bressuire

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more

          Cabo Verde

          Image
            Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
          Read more

          Gyllenstierna

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more