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

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

Commented by bemath last updated on 08/Jun/20

$$\mathrm{for}\mathrm{xy}=-15\mathrm{and}\mathrm{z}=10$$$$\mathrm{then}\max \mathrm{of}\mathrm{w}=\mathrm{z}-\mathrm{xy}=10-(-15)=25$$

Answered by 1549442205 last updated on 08/Jun/20

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

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