Find $p$ and $q$ such that the maximum and minimum values of $5 +6costheta + 2 cos2theta$ satisfy $x^2-p x+q=2$

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If the maximum and minimum values of $5 +6costheta + 2 cos2theta$ satisfy $x^2-px + q = 2$, then what are $p$ and $q$?




My thinking that, maximum valus of $costheta$ is $1$ and minimum is $-1$. Now I can claim that the maximum of $5+6costheta + 2 cos2theta)$ is $13$ and the minimum value is $9$



So $p = 13$ and $q=9$.



Is it correct or not?



Any hints/solution?



Thank u




real-analysis




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  • 1




    Where did you get your numbers? If we take $theta = pi$ then your expression is $5-6+2=1$ so $9$ certainly isn't the minimum.
    – lulu
    Jul 31 at 0:20






  • 1




    Min seems like 3/4.
    – Mason
    Jul 31 at 0:22










  • yaa,,,that was my mistake @lulu
    – Messi fifa
    Jul 31 at 0:22






  • 1




    Use $ cos(2 theta) =2 cos^2 theta -1$ and complete the square ... Now have a little think about what the maximum & minimum values will be.
    – Donald Splutterwit
    Jul 31 at 0:23






  • 1




    $p$ will be the sum of the maximum & minimum. $q-2$ will be their product.
    – Donald Splutterwit
    Jul 31 at 0:52














up vote
1
down vote

favorite
1













If the maximum and minimum values of $5 +6costheta + 2 cos2theta$ satisfy $x^2-px + q = 2$, then what are $p$ and $q$?




My thinking that, maximum valus of $costheta$ is $1$ and minimum is $-1$. Now I can claim that the maximum of $5+6costheta + 2 cos2theta)$ is $13$ and the minimum value is $9$



So $p = 13$ and $q=9$.



Is it correct or not?



Any hints/solution?



Thank u




real-analysis




share|cite|improve this question

















  • 1




    Where did you get your numbers? If we take $theta = pi$ then your expression is $5-6+2=1$ so $9$ certainly isn't the minimum.
    – lulu
    Jul 31 at 0:20






  • 1




    Min seems like 3/4.
    – Mason
    Jul 31 at 0:22










  • yaa,,,that was my mistake @lulu
    – Messi fifa
    Jul 31 at 0:22






  • 1




    Use $ cos(2 theta) =2 cos^2 theta -1$ and complete the square ... Now have a little think about what the maximum & minimum values will be.
    – Donald Splutterwit
    Jul 31 at 0:23






  • 1




    $p$ will be the sum of the maximum & minimum. $q-2$ will be their product.
    – Donald Splutterwit
    Jul 31 at 0:52












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1






If the maximum and minimum values of $5 +6costheta + 2 cos2theta$ satisfy $x^2-px + q = 2$, then what are $p$ and $q$?




My thinking that, maximum valus of $costheta$ is $1$ and minimum is $-1$. Now I can claim that the maximum of $5+6costheta + 2 cos2theta)$ is $13$ and the minimum value is $9$



So $p = 13$ and $q=9$.



Is it correct or not?



Any hints/solution?



Thank u




real-analysis




share|cite|improve this question














If the maximum and minimum values of $5 +6costheta + 2 cos2theta$ satisfy $x^2-px + q = 2$, then what are $p$ and $q$?




My thinking that, maximum valus of $costheta$ is $1$ and minimum is $-1$. Now I can claim that the maximum of $5+6costheta + 2 cos2theta)$ is $13$ and the minimum value is $9$



So $p = 13$ and $q=9$.



Is it correct or not?



Any hints/solution?



Thank u





real-analysis





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 31 at 1:34









Blue

43.6k868141




43.6k868141









asked Jul 31 at 0:15









Messi fifa

1478




1478







  • 1




    Where did you get your numbers? If we take $theta = pi$ then your expression is $5-6+2=1$ so $9$ certainly isn't the minimum.
    – lulu
    Jul 31 at 0:20






  • 1




    Min seems like 3/4.
    – Mason
    Jul 31 at 0:22










  • yaa,,,that was my mistake @lulu
    – Messi fifa
    Jul 31 at 0:22






  • 1




    Use $ cos(2 theta) =2 cos^2 theta -1$ and complete the square ... Now have a little think about what the maximum & minimum values will be.
    – Donald Splutterwit
    Jul 31 at 0:23






  • 1




    $p$ will be the sum of the maximum & minimum. $q-2$ will be their product.
    – Donald Splutterwit
    Jul 31 at 0:52












  • 1




    Where did you get your numbers? If we take $theta = pi$ then your expression is $5-6+2=1$ so $9$ certainly isn't the minimum.
    – lulu
    Jul 31 at 0:20






  • 1




    Min seems like 3/4.
    – Mason
    Jul 31 at 0:22










  • yaa,,,that was my mistake @lulu
    – Messi fifa
    Jul 31 at 0:22






  • 1




    Use $ cos(2 theta) =2 cos^2 theta -1$ and complete the square ... Now have a little think about what the maximum & minimum values will be.
    – Donald Splutterwit
    Jul 31 at 0:23






  • 1




    $p$ will be the sum of the maximum & minimum. $q-2$ will be their product.
    – Donald Splutterwit
    Jul 31 at 0:52







1




1




Where did you get your numbers? If we take $theta = pi$ then your expression is $5-6+2=1$ so $9$ certainly isn't the minimum.
– lulu
Jul 31 at 0:20




Where did you get your numbers? If we take $theta = pi$ then your expression is $5-6+2=1$ so $9$ certainly isn't the minimum.
– lulu
Jul 31 at 0:20




1




1




Min seems like 3/4.
– Mason
Jul 31 at 0:22




Min seems like 3/4.
– Mason
Jul 31 at 0:22












yaa,,,that was my mistake @lulu
– Messi fifa
Jul 31 at 0:22




yaa,,,that was my mistake @lulu
– Messi fifa
Jul 31 at 0:22




1




1




Use $ cos(2 theta) =2 cos^2 theta -1$ and complete the square ... Now have a little think about what the maximum & minimum values will be.
– Donald Splutterwit
Jul 31 at 0:23




Use $ cos(2 theta) =2 cos^2 theta -1$ and complete the square ... Now have a little think about what the maximum & minimum values will be.
– Donald Splutterwit
Jul 31 at 0:23




1




1




$p$ will be the sum of the maximum & minimum. $q-2$ will be their product.
– Donald Splutterwit
Jul 31 at 0:52




$p$ will be the sum of the maximum & minimum. $q-2$ will be their product.
– Donald Splutterwit
Jul 31 at 0:52










1 Answer
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The minimum value is wrong. Let $f(t)=5+6t+2(2t^2-1)$. [Recall that $cos (2x)=2cos^2x-1$]. Therefore, $f'(t)=6+8t=0$ when $t=-3/4$ which is a possible value for $cos x$. So the minimum value is $5+6(-3/4)+2(9/8-1)=3/4$. The quadratic with leading coefficient $1$ and roots $3/4$ and $13$ is given by $(x-frac 3 4 ) (x-13)=0$. Comparing coefficienrs we get $p=frac 55 4$ adn $q=frac 47 4$






share|cite|improve this answer



















  • 1




    then what is the value of p and q ?
    – Messi fifa
    Jul 31 at 0:37






  • 1




    @Messififa I have edited my answer.
    – Kavi Rama Murthy
    Jul 31 at 5:32










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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
2
down vote



accepted










The minimum value is wrong. Let $f(t)=5+6t+2(2t^2-1)$. [Recall that $cos (2x)=2cos^2x-1$]. Therefore, $f'(t)=6+8t=0$ when $t=-3/4$ which is a possible value for $cos x$. So the minimum value is $5+6(-3/4)+2(9/8-1)=3/4$. The quadratic with leading coefficient $1$ and roots $3/4$ and $13$ is given by $(x-frac 3 4 ) (x-13)=0$. Comparing coefficienrs we get $p=frac 55 4$ adn $q=frac 47 4$






share|cite|improve this answer



















  • 1




    then what is the value of p and q ?
    – Messi fifa
    Jul 31 at 0:37






  • 1




    @Messififa I have edited my answer.
    – Kavi Rama Murthy
    Jul 31 at 5:32














up vote
2
down vote



accepted










The minimum value is wrong. Let $f(t)=5+6t+2(2t^2-1)$. [Recall that $cos (2x)=2cos^2x-1$]. Therefore, $f'(t)=6+8t=0$ when $t=-3/4$ which is a possible value for $cos x$. So the minimum value is $5+6(-3/4)+2(9/8-1)=3/4$. The quadratic with leading coefficient $1$ and roots $3/4$ and $13$ is given by $(x-frac 3 4 ) (x-13)=0$. Comparing coefficienrs we get $p=frac 55 4$ adn $q=frac 47 4$






share|cite|improve this answer



















  • 1




    then what is the value of p and q ?
    – Messi fifa
    Jul 31 at 0:37






  • 1




    @Messififa I have edited my answer.
    – Kavi Rama Murthy
    Jul 31 at 5:32












up vote
2
down vote



accepted







up vote
2
down vote



accepted






The minimum value is wrong. Let $f(t)=5+6t+2(2t^2-1)$. [Recall that $cos (2x)=2cos^2x-1$]. Therefore, $f'(t)=6+8t=0$ when $t=-3/4$ which is a possible value for $cos x$. So the minimum value is $5+6(-3/4)+2(9/8-1)=3/4$. The quadratic with leading coefficient $1$ and roots $3/4$ and $13$ is given by $(x-frac 3 4 ) (x-13)=0$. Comparing coefficienrs we get $p=frac 55 4$ adn $q=frac 47 4$






share|cite|improve this answer















The minimum value is wrong. Let $f(t)=5+6t+2(2t^2-1)$. [Recall that $cos (2x)=2cos^2x-1$]. Therefore, $f'(t)=6+8t=0$ when $t=-3/4$ which is a possible value for $cos x$. So the minimum value is $5+6(-3/4)+2(9/8-1)=3/4$. The quadratic with leading coefficient $1$ and roots $3/4$ and $13$ is given by $(x-frac 3 4 ) (x-13)=0$. Comparing coefficienrs we get $p=frac 55 4$ adn $q=frac 47 4$







share|cite|improve this answer















share|cite|improve this answer



share|cite|improve this answer








edited Jul 31 at 5:31


























answered Jul 31 at 0:22









Kavi Rama Murthy

19.6k2829




19.6k2829







  • 1




    then what is the value of p and q ?
    – Messi fifa
    Jul 31 at 0:37






  • 1




    @Messififa I have edited my answer.
    – Kavi Rama Murthy
    Jul 31 at 5:32












  • 1




    then what is the value of p and q ?
    – Messi fifa
    Jul 31 at 0:37






  • 1




    @Messififa I have edited my answer.
    – Kavi Rama Murthy
    Jul 31 at 5:32







1




1




then what is the value of p and q ?
– Messi fifa
Jul 31 at 0:37




then what is the value of p and q ?
– Messi fifa
Jul 31 at 0:37




1




1




@Messififa I have edited my answer.
– Kavi Rama Murthy
Jul 31 at 5:32




@Messififa I have edited my answer.
– Kavi Rama Murthy
Jul 31 at 5:32












 

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