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Question Number 136186 by ZiYangLee last updated on 19/Mar/21

Let f(x)=cos^(−1) x+cos^(−1) (x−1), find the range of f.

$$\mathrm{Let}\:{f}\left({x}\right)=\mathrm{cos}^{−\mathrm{1}} {x}+\mathrm{cos}^{−\mathrm{1}} \left({x}−\mathrm{1}\right),\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{f}. \\ $$

Answered by ZiYangLee last updated on 19/Mar/21

Ans: [(π/2),((3π)/2)]

$$\mathrm{Ans}:\:\left[\frac{\pi}{\mathrm{2}},\frac{\mathrm{3}\pi}{\mathrm{2}}\right] \\ $$

Answered by MJS_new last updated on 19/Mar/21

f(x) is defined for 0≤x≤1 and is strictly monotonically decreasing ⇒ f(1)≤f(x)≤f(0) ⇔ (π/2)≤f(x)≤((3π)/2)

$${f}\left({x}\right)\:\mathrm{is}\:\mathrm{defined}\:\mathrm{for}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:\mathrm{and}\:\mathrm{is}\:\mathrm{strictly} \\ $$$$\mathrm{monotonically}\:\mathrm{decreasing} \\ $$$$\Rightarrow \\ $$$${f}\left(\mathrm{1}\right)\leqslant{f}\left({x}\right)\leqslant{f}\left(\mathrm{0}\right)\:\Leftrightarrow\:\frac{\pi}{\mathrm{2}}\leqslant{f}\left({x}\right)\leqslant\frac{\mathrm{3}\pi}{\mathrm{2}} \\ $$

Commented by ZiYangLee last updated on 20/Mar/21

thanks

$$\mathrm{thanks} \\ $$

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