Question for math lovers-2

Algebra Level 3

Given x,y $x,y\in R,\quad { x }^{ 2 }+{ y }^{ 2 }>0$ . If the maximum and minimum value of the expression $E=\frac { { x }^{ 2 }+{ y }^{ 2 } }{ { x }^{ 2 }+xy+4y^2 }$ are M and m, compute $\frac{ 2007}{2} \times (M + m )$ .

1338 845 1453 1101

1 solution

Patrick Corn
Jun 30, 2014

I'm assuming that $4y$ should be a $4y^2$ . In this case, if $y = 0$ we get $E = 1$ . Otherwise, we can set $r= x/y$ and divide through to get $E= \frac{r^2+1}{r^2+r+4}$ . Calculus gives critical points of $r = -3 \pm \sqrt{10}$ ; we can check that these give a global min and max respectively. Plugging in and adding gives $M + m = 4/3$ , so the answer is $\fbox{1338}$ .

@Kandarp Singh Please check the equation. As stated by Patrick and others, the denominator should be $4y^2$ as opposed to $4y$ . The question is not "perfectly OK".

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Calvin Lin Staff - 6 years, 11 months ago

doubt please help suppose x^2 +y^2 = m^2 therefore dividing the fraction by m^2 then it becomes 2/( 5 + 3cos2a +sin2a) where sina = x/m therefore its minimum value is obtained when denominator is maximum sina + cosa maximum value is root 2 therefore angle pi/8 and thus we get minimum value as 2 . 2^(1/2) /( 5 . 2^(1/2) + 4) and for maximum value angle =pi/8 = 2 . 2^(1/2) /( 5 . 2^(1/2) - 4) the answer comes as 1180.588

U Z - 6 years, 8 months ago

Question for math lovers-2

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