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Question Number 138235 by mr W last updated on 11/Apr/21
for p,q∈R satisfying p^4 +q^4 =4pq  find the range of p+q when  1) no restriction  2) 0≤p≤1, 0≤q≤1
$${for}\:{p},{q}\in\mathbb{R}\:{satisfying}\:{p}^{\mathrm{4}} +{q}^{\mathrm{4}} =\mathrm{4}{pq} \\ $$$${find}\:{the}\:{range}\:{of}\:{p}+{q}\:{when} \\ $$$$\left.\mathrm{1}\right)\:{no}\:{restriction} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{0}\leqslant{p}\leqslant\mathrm{1},\:\mathrm{0}\leqslant{q}\leqslant\mathrm{1} \\ $$
Answered by mr W last updated on 12/Apr/21
Commented by mr W last updated on 12/Apr/21
p+q=k  1) without restriction  k_(max) =2(√2)  k_(min) =−2(√2)  2) with restriction: 0≤p,q≤1  k_(max) ≈1.25099  k_(min) =0
$${p}+{q}={k} \\ $$$$\left.\mathrm{1}\right)\:{without}\:{restriction} \\ $$$${k}_{{max}} =\mathrm{2}\sqrt{\mathrm{2}} \\ $$$${k}_{{min}} =−\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{with}\:{restriction}:\:\mathrm{0}\leqslant{p},{q}\leqslant\mathrm{1} \\ $$$${k}_{{max}} \approx\mathrm{1}.\mathrm{25099} \\ $$$${k}_{{min}} =\mathrm{0} \\ $$

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