Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 65868 by aliesam last updated on 05/Aug/19

Commented by MJS last updated on 05/Aug/19

examine the function f(x)=(((3x−4)(5x+4))/((3x−5)(x+5))) it′s defined for x∈R\{−5; (5/3)} local maximum at ((((109−(√(4921)))/(58))),(((383−3(√(4921)))/(200))) ) ≈ (((.67)),((.86)) ) local minimum at ((((109+(√(4921)))/(58))),(((383+3(√(4921)))/(200))) ) ≈ (((3.09)),((2.97)) ) we have to try 0<n<(5/3) ⇒ n=1 and n>(5/3) ⇒ n≥2 but lim_(n→∞) f(x)=5 ⇒ we′re looking for f(x)∈{3; 4} ⇒ the only solutions are n=2 ⇒ f(n)=4 n=14 ⇒ f(n)=4

$$\mathrm{examine}\:\mathrm{the}\:\mathrm{function} \\ $$$${f}\left({x}\right)=\frac{\left(\mathrm{3}{x}−\mathrm{4}\right)\left(\mathrm{5}{x}+\mathrm{4}\right)}{\left(\mathrm{3}{x}−\mathrm{5}\right)\left({x}+\mathrm{5}\right)} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{defined}\:\mathrm{for}\:{x}\in\mathbb{R}\backslash\left\{−\mathrm{5};\:\frac{\mathrm{5}}{\mathrm{3}}\right\} \\ $$$$\mathrm{local}\:\mathrm{maximum}\:\mathrm{at}\:\begin{pmatrix}{\frac{\mathrm{109}−\sqrt{\mathrm{4921}}}{\mathrm{58}}}\\{\frac{\mathrm{383}−\mathrm{3}\sqrt{\mathrm{4921}}}{\mathrm{200}}}\end{pmatrix}\:\approx\begin{pmatrix}{.\mathrm{67}}\\{.\mathrm{86}}\end{pmatrix} \\ $$$$\mathrm{local}\:\mathrm{minimum}\:\mathrm{at}\:\begin{pmatrix}{\frac{\mathrm{109}+\sqrt{\mathrm{4921}}}{\mathrm{58}}}\\{\frac{\mathrm{383}+\mathrm{3}\sqrt{\mathrm{4921}}}{\mathrm{200}}}\end{pmatrix}\:\approx\begin{pmatrix}{\mathrm{3}.\mathrm{09}}\\{\mathrm{2}.\mathrm{97}}\end{pmatrix} \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{to}\:\mathrm{try}\:\mathrm{0}<{n}<\frac{\mathrm{5}}{\mathrm{3}}\:\Rightarrow\:{n}=\mathrm{1} \\ $$$$\mathrm{and}\:{n}>\frac{\mathrm{5}}{\mathrm{3}}\:\Rightarrow\:{n}\geqslant\mathrm{2} \\ $$$$\mathrm{but} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{f}\left({x}\right)=\mathrm{5}\:\Rightarrow\:\mathrm{we}'\mathrm{re}\:\mathrm{looking}\:\mathrm{for}\:{f}\left({x}\right)\in\left\{\mathrm{3};\:\mathrm{4}\right\} \\ $$$$\Rightarrow\:\mathrm{the}\:\mathrm{only}\:\mathrm{solutions}\:\mathrm{are} \\ $$$${n}=\mathrm{2}\:\Rightarrow\:{f}\left({n}\right)=\mathrm{4} \\ $$$${n}=\mathrm{14}\:\Rightarrow\:{f}\left({n}\right)=\mathrm{4} \\ $$

Commented by MJS last updated on 05/Aug/19

same procedure shows that there′s no solution for n≤0

$$\mathrm{same}\:\mathrm{procedure}\:\mathrm{shows}\:\mathrm{that}\:\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{solution} \\ $$$$\mathrm{for}\:{n}\leqslant\mathrm{0} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com