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Question Number 151100 by mathdanisur last updated on 18/Aug/21

$$\mathrm{if}0⩽\mathrm{x};\mathrm{y};\mathrm{z}⩽\mathrm{k}\mathrm{and}\mathrm{k}>0\mathrm{then}:$$ $$\mathrm{y}(\mathrm{x}-\mathrm{z})-\mathrm{z}(\mathrm{x}-\mathrm{k})⩽{\mathrm{k}}^2$$

Answered by dumitrel last updated on 18/Aug/21

$$\iff y(x-z)+z(k-x)⩽k^2$$ $$I.Ifx⩽z\Rightarrow y(x-z)⩽0$$ $$z(k-x)⩽k\cdot k=k^2\Rightarrow$$ $$\Rightarrow y(x-z)+z(k-x)⩽k^2$$ $$II.Ifx>z\Rightarrow$$ $$\mid y(x-z)+z(k-x)\mid ⩽\mid y(x-z)\mid +\mid z(k-x)\mid ⩽$$ $$⩽k(x-z)+k(k-x)=k(k-z)⩽k^2$$ $$\Rightarrow y(x-z)+z(k-x)⩽k^2$$ $$$$ $$$$ $$$$

Commented bymathdanisur last updated on 18/Aug/21

$$\mathrm{nice}\mathbf{S}\mathrm{er}\mathrm{thank}\mathrm{you}$$

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