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sketch-the-graph-of-f-x-2-x-x-2-then-state-its-domain-and-range-




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

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