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Minimum of $left(a + b + c + dright)left(frac1a + frac1b + frac4c + frac16dright)$
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If $a$, $b$, $c$, $d$ are positive integers, find the minimum value of
$$P = left(a + b + c + dright)left(frac1a + frac1b + frac4c + frac16dright)$$
and the values of $a$, $b$, $c$, $d$ when it is reached.
My try:
$$left.
beginarrayl
a + b + c + d ge 4sqrt[4]abcd\
frac1a + frac1b + frac4c + frac16d ge 4sqrt[4]frac64abcd
endarray
right}
Rightarrow P ge 32sqrt2$$
I have used mean inequalities, but that doesn't mean that I have found the minimum value. Also, I have found a similar exercise here (exercise #5), but the author shows that $P ge 64$, which is greater than what have I found.
Can you help me solve the problem, please? Thanks!
inequality
asked Jul 19 at 10:24
Iulian Oleniuc
3619
add a comment |Â
up vote
1
down vote
favorite
If $a$, $b$, $c$, $d$ are positive integers, find the minimum value of
$$P = left(a + b + c + dright)left(frac1a + frac1b + frac4c + frac16dright)$$
and the values of $a$, $b$, $c$, $d$ when it is reached.
My try:
$$left.
beginarrayl
a + b + c + d ge 4sqrt[4]abcd\
frac1a + frac1b + frac4c + frac16d ge 4sqrt[4]frac64abcd
endarray
right}
Rightarrow P ge 32sqrt2$$
I have used mean inequalities, but that doesn't mean that I have found the minimum value. Also, I have found a similar exercise here (exercise #5), but the author shows that $P ge 64$, which is greater than what have I found.
Can you help me solve the problem, please? Thanks!
inequality
asked Jul 19 at 10:24
Iulian Oleniuc
3619
Hint: Use CSB inequality. (As pointed out in the hyperlinked article.)
– Jose Arnaldo Bebita Dris
Jul 19 at 10:29
@JoseArnaldoBebitaDris, I have found a better value than the one with CSB.
– Iulian Oleniuc
Jul 19 at 10:33
2
@IulianOleniuc Your lower bound is not sharp. So, it is not really better.
– Batominovski
Jul 19 at 10:39
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If $a$, $b$, $c$, $d$ are positive integers, find the minimum value of
$$P = left(a + b + c + dright)left(frac1a + frac1b + frac4c + frac16dright)$$
and the values of $a$, $b$, $c$, $d$ when it is reached.
My try:
$$left.
beginarrayl
a + b + c + d ge 4sqrt[4]abcd\
frac1a + frac1b + frac4c + frac16d ge 4sqrt[4]frac64abcd
endarray
right}
Rightarrow P ge 32sqrt2$$
I have used mean inequalities, but that doesn't mean that I have found the minimum value. Also, I have found a similar exercise here (exercise #5), but the author shows that $P ge 64$, which is greater than what have I found.
Can you help me solve the problem, please? Thanks!
inequality
asked Jul 19 at 10:24
Iulian Oleniuc
3619
If $a$, $b$, $c$, $d$ are positive integers, find the minimum value of
$$P = left(a + b + c + dright)left(frac1a + frac1b + frac4c + frac16dright)$$
and the values of $a$, $b$, $c$, $d$ when it is reached.
My try:
$$left.
beginarrayl
a + b + c + d ge 4sqrt[4]abcd\
frac1a + frac1b + frac4c + frac16d ge 4sqrt[4]frac64abcd
endarray
right}
Rightarrow P ge 32sqrt2$$
I have used mean inequalities, but that doesn't mean that I have found the minimum value. Also, I have found a similar exercise here (exercise #5), but the author shows that $P ge 64$, which is greater than what have I found.
Can you help me solve the problem, please? Thanks!
inequality
asked Jul 19 at 10:24
Iulian Oleniuc
3619
asked Jul 19 at 10:24
Iulian Oleniuc
3619
asked Jul 19 at 10:24
Iulian Oleniuc
3619
asked Jul 19 at 10:24
Iulian Oleniuc
3619
3619
Hint: Use CSB inequality. (As pointed out in the hyperlinked article.)
– Jose Arnaldo Bebita Dris
Jul 19 at 10:29
@JoseArnaldoBebitaDris, I have found a better value than the one with CSB.
– Iulian Oleniuc
Jul 19 at 10:33
2
@IulianOleniuc Your lower bound is not sharp. So, it is not really better.
– Batominovski
Jul 19 at 10:39
add a comment |Â
Hint: Use CSB inequality. (As pointed out in the hyperlinked article.)
– Jose Arnaldo Bebita Dris
Jul 19 at 10:29
@JoseArnaldoBebitaDris, I have found a better value than the one with CSB.
– Iulian Oleniuc
Jul 19 at 10:33
2
@IulianOleniuc Your lower bound is not sharp. So, it is not really better.
– Batominovski
Jul 19 at 10:39
Hint: Use CSB inequality. (As pointed out in the hyperlinked article.)
– Jose Arnaldo Bebita Dris
Jul 19 at 10:29
Hint: Use CSB inequality. (As pointed out in the hyperlinked article.)
– Jose Arnaldo Bebita Dris
Jul 19 at 10:29
@JoseArnaldoBebitaDris, I have found a better value than the one with CSB.
– Iulian Oleniuc
Jul 19 at 10:33
@JoseArnaldoBebitaDris, I have found a better value than the one with CSB.
– Iulian Oleniuc
Jul 19 at 10:33
2
2
@IulianOleniuc Your lower bound is not sharp. So, it is not really better.
– Batominovski
Jul 19 at 10:39
@IulianOleniuc Your lower bound is not sharp. So, it is not really better.
– Batominovski
Jul 19 at 10:39
add a comment |Â
1 Answer
1
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oldest
votes
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2
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accepted
If you want to use the AM-GM Inequality, it can be done as follows. Observe that
$$a+b+c+d=a+b+2left(fracc2right)+4left(fracd4right)geq 8sqrt[8]ableft(fracc2right)^2left(fracd4right)^4$$
and that
$$frac1a+frac1b+frac4c+frac16d=frac1a+frac1b+2left(frac2cright)+4left(frac4dright)geq 8sqrt[8]left(frac1aright)left(frac1bright)left(frac2cright)^2left(frac4dright)^4,.$$
However, using the Cauchy-Schwarz Inequality is probably the easiest way. (The equality holds iff there exists $lambda >0$ such that $(a,b,c,d)=(lambda,lambda,2lambda,4lambda)$.)
answered Jul 19 at 10:36
Batominovski
23.2k22777
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
If you want to use the AM-GM Inequality, it can be done as follows. Observe that
$$a+b+c+d=a+b+2left(fracc2right)+4left(fracd4right)geq 8sqrt[8]ableft(fracc2right)^2left(fracd4right)^4$$
and that
$$frac1a+frac1b+frac4c+frac16d=frac1a+frac1b+2left(frac2cright)+4left(frac4dright)geq 8sqrt[8]left(frac1aright)left(frac1bright)left(frac2cright)^2left(frac4dright)^4,.$$
However, using the Cauchy-Schwarz Inequality is probably the easiest way. (The equality holds iff there exists $lambda >0$ such that $(a,b,c,d)=(lambda,lambda,2lambda,4lambda)$.)
answered Jul 19 at 10:36
Batominovski
23.2k22777
add a comment |Â
up vote
2
down vote
accepted
If you want to use the AM-GM Inequality, it can be done as follows. Observe that
$$a+b+c+d=a+b+2left(fracc2right)+4left(fracd4right)geq 8sqrt[8]ableft(fracc2right)^2left(fracd4right)^4$$
and that
$$frac1a+frac1b+frac4c+frac16d=frac1a+frac1b+2left(frac2cright)+4left(frac4dright)geq 8sqrt[8]left(frac1aright)left(frac1bright)left(frac2cright)^2left(frac4dright)^4,.$$
However, using the Cauchy-Schwarz Inequality is probably the easiest way. (The equality holds iff there exists $lambda >0$ such that $(a,b,c,d)=(lambda,lambda,2lambda,4lambda)$.)
answered Jul 19 at 10:36
Batominovski
23.2k22777
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
If you want to use the AM-GM Inequality, it can be done as follows. Observe that
$$a+b+c+d=a+b+2left(fracc2right)+4left(fracd4right)geq 8sqrt[8]ableft(fracc2right)^2left(fracd4right)^4$$
and that
$$frac1a+frac1b+frac4c+frac16d=frac1a+frac1b+2left(frac2cright)+4left(frac4dright)geq 8sqrt[8]left(frac1aright)left(frac1bright)left(frac2cright)^2left(frac4dright)^4,.$$
However, using the Cauchy-Schwarz Inequality is probably the easiest way. (The equality holds iff there exists $lambda >0$ such that $(a,b,c,d)=(lambda,lambda,2lambda,4lambda)$.)
answered Jul 19 at 10:36
Batominovski
23.2k22777
If you want to use the AM-GM Inequality, it can be done as follows. Observe that
$$a+b+c+d=a+b+2left(fracc2right)+4left(fracd4right)geq 8sqrt[8]ableft(fracc2right)^2left(fracd4right)^4$$
and that
$$frac1a+frac1b+frac4c+frac16d=frac1a+frac1b+2left(frac2cright)+4left(frac4dright)geq 8sqrt[8]left(frac1aright)left(frac1bright)left(frac2cright)^2left(frac4dright)^4,.$$
However, using the Cauchy-Schwarz Inequality is probably the easiest way. (The equality holds iff there exists $lambda >0$ such that $(a,b,c,d)=(lambda,lambda,2lambda,4lambda)$.)
answered Jul 19 at 10:36
Batominovski
23.2k22777
edited Jul 19 at 10:41
answered Jul 19 at 10:36
Batominovski
23.2k22777
answered Jul 19 at 10:36
Batominovski
23.2k22777
answered Jul 19 at 10:36
Batominovski
23.2k22777
23.2k22777
add a comment |Â
add a comment |Â
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Hint: Use CSB inequality. (As pointed out in the hyperlinked article.)
– Jose Arnaldo Bebita Dris
Jul 19 at 10:29
@JoseArnaldoBebitaDris, I have found a better value than the one with CSB.
– Iulian Oleniuc
Jul 19 at 10:33
2
@IulianOleniuc Your lower bound is not sharp. So, it is not really better.
– Batominovski
Jul 19 at 10:39