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Question Number 97438 by john santu last updated on 08/Jun/20

If −3≤x≤4, −2≤y≤5, 4≤z≤10 , find the greatest value of w = z−xy

$$\mathrm{If}\:−\mathrm{3}\leqslant\mathrm{x}\leqslant\mathrm{4},\:−\mathrm{2}\leqslant\mathrm{y}\leqslant\mathrm{5},\:\mathrm{4}\leqslant\mathrm{z}\leqslant\mathrm{10} \\ $$$$,\:\mathrm{find}\:\mathrm{the}\:\mathrm{greatest} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{w}\:=\:\mathrm{z}−\mathrm{xy}\: \\ $$

Commented by bemath last updated on 08/Jun/20

for xy = −15 and z = 10 then max of w = z−xy = 10−(−15)=25

$$\mathrm{for}\:\mathrm{xy}\:=\:−\mathrm{15}\:\mathrm{and}\:\mathrm{z}\:=\:\mathrm{10} \\ $$$$\mathrm{then}\:\mathrm{max}\:\mathrm{of}\:\mathrm{w}\:=\:\mathrm{z}−\mathrm{xy}\:=\:\mathrm{10}−\left(−\mathrm{15}\right)=\mathrm{25} \\ $$

Answered by 1549442205 last updated on 08/Jun/20

From the hypothesis { ((−3≤x≤4)),((−2≤y≤5)) :} we get 20≥xy≥−15⇒−20≤−xy≤15(1)which by the combine to 4≤z≤10 (2),plus two same dimension inequalities (1)and (2) we obtain that −16≤z−xy≤25.This means z−xy_(min) =−16 when (x;y;z)=(4;5;4)and z−xy_(max) =25 when (x;y;z)=(−3;5;10)

$$\mathrm{From}\:\mathrm{the}\:\mathrm{hypothesis}\:\begin{cases}{−\mathrm{3}\leqslant\mathrm{x}\leqslant\mathrm{4}}\\{−\mathrm{2}\leqslant\mathrm{y}\leqslant\mathrm{5}}\end{cases}\:\mathrm{we}\:\mathrm{get} \\ $$$$\mathrm{20}\geqslant\mathrm{xy}\geqslant−\mathrm{15}\Rightarrow−\mathrm{20}\leqslant−\mathrm{xy}\leqslant\mathrm{15}\left(\mathrm{1}\right)\mathrm{which} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{combine}\:\mathrm{to}\:\mathrm{4}\leqslant\mathrm{z}\leqslant\mathrm{10}\:\left(\mathrm{2}\right),\mathrm{plus}\:\mathrm{two}\:\mathrm{same}\:\mathrm{dimension}\:\mathrm{inequalities}\: \\ $$$$\left(\mathrm{1}\right)\mathrm{and}\:\left(\mathrm{2}\right)\:\mathrm{we}\:\mathrm{obtain}\:\mathrm{that}\:−\mathrm{16}\leqslant\mathrm{z}−\mathrm{xy}\leqslant\mathrm{25}.\mathrm{This}\:\mathrm{means} \\ $$$$\mathrm{z}−\mathrm{xy}_{\mathrm{min}} =−\mathrm{16}\:\mathrm{when}\:\left(\mathrm{x};\mathrm{y};\mathrm{z}\right)=\left(\mathrm{4};\mathrm{5};\mathrm{4}\right)\mathrm{and} \\ $$$$\mathrm{z}−\mathrm{xy}_{\mathrm{max}} =\mathrm{25}\:\mathrm{when}\:\left(\mathrm{x};\mathrm{y};\mathrm{z}\right)=\left(−\mathrm{3};\mathrm{5};\mathrm{10}\right) \\ $$

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