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Question Number 205490 by mnjuly1970 last updated on 22/Mar/24

$$$$$$If,f\left(x\right)=\sqrt{2+x}+a\sqrt{x-1}$$$$ismonotonefunction.$$$$findtherangeof{}^{″}a{}^{″}$$$$$$

Answered by mr W last updated on 22/Mar/24

$$x⩾1$$$$f\left(x\right)=\sqrt{3+x-1}+a\sqrt{x-1}$$$$lett=x-1⩾0$$$$f\left(t\right)=\sqrt{3+t^2}+at$$$$f^{\prime }\left(t\right)=\frac{t}{\sqrt{3+t^2}}+a=\frac{1}{\sqrt{1+\frac{3}{t^2}}}+a$$$$suchthatthefunctionismonotone$$$$fort⩾0,\frac{1}{\sqrt{1+\frac{3}{t^2}}}+a=0shouldhave$$$$nosolution.thisisthecase:$$$$1)ifa⩾0$$$$2)ifa⩽-1$$$$i.e.therangeofaisa⩾0ora⩽-1$$$$or$$$$functionisnotmonotoneif$$$$-1<a<0$$

Commented by mnjuly1970 last updated on 22/Mar/24

$$thanksalotsir...$$

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