Question Number 35645 by mondodotto@gmail.com last updated on 21/May/18
$$\boldsymbol{\mathrm{sketch}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{graph}}\:\boldsymbol{\mathrm{of}} \\ $$$$\boldsymbol{\mathrm{f}}\left(\boldsymbol{{x}}\right)=\mathrm{2}−\boldsymbol{{x}}−\boldsymbol{{x}}^{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{then}}\:\boldsymbol{\mathrm{state}}\:\boldsymbol{\mathrm{its}}\:\boldsymbol{\mathrm{domain}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{range}} \\ $$
Answered by Penguin last updated on 21/May/18
![Domain: x∈[−∞,∞] Range: Max min y @ y′=0 y′=−1−2x=0 x=−(1/2) y=2−(−(1/2))−(−(1/2))^2 y=2+(1/2)−(1/4) y=(9/4) max/min? y′′=−2 ∴Maximum ∴y∈[−∞, (9/4)]](https://www.tinkutara.com/question/Q35679.png)
$$\mathrm{Domain}: \\ $$$${x}\in\left[−\infty,\infty\right] \\ $$$$\: \\ $$$$\mathrm{Range}: \\ $$$$\mathrm{Max}\:\mathrm{min}\:{y}\:@\:{y}'=\mathrm{0} \\ $$$${y}'=−\mathrm{1}−\mathrm{2}{x}=\mathrm{0} \\ $$$${x}=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${y}=\mathrm{2}−\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)−\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$${y}=\mathrm{2}+\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${y}=\frac{\mathrm{9}}{\mathrm{4}} \\ $$$$\: \\ $$$$\mathrm{max}/\mathrm{min}? \\ $$$${y}''=−\mathrm{2} \\ $$$$\therefore\mathrm{Maximum} \\ $$$$\: \\ $$$$\therefore{y}\in\left[−\infty,\:\frac{\mathrm{9}}{\mathrm{4}}\right] \\ $$