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If the equation x^(4)+px^(3)+qx^(2)+rx+5...

If the equation `x^(4)+px^(3)+qx^(2)+rx+5=0` has four positive real roots, find the maximum value of `pr`.

Text Solution

Verified by Experts

The correct Answer is:
80
`pr(-p)(-r)`
`=(alpha+beta+gamma+delta)(alpha beta gamma+alpha beta delta+gamma delta alpha+gamma delta beta)`
`=alpha^(2) beta gamma + alpha^(2) beta delta +alpha^(2) gamma delta+alpha beta gamma delta+beta^(2) gamma alpha`
`+beta^(2) alpha delta +alpha beta gamma delta+beta^(2) gamma delta +gamma^(2) alpha beta+alpha beta gamma delta`
`+gamma^(2) delta alpha +gamma^(2) delta beta+alpha beta gamma delta +alpha beta delta^(2)+gamma alpha delta^(2) + gamm beta delta^(2)`
`:' AM ge GM`
`implies(pr)/16ge(alpha^(16)beta^(16)gamma^(16)deltal^(`6))^(1/6)=alpha beta gamma delta=5`
`implies(pr)/16ge5`
or `prge 80`
`:.` Minimum value of `pr` ils 80.
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