Question

Sa se studieze monotonia functiei f:(-3;3)->R, f(x)= $ln\frac{3+x}{3-x}$

Answer 1

Foloseste prima derivata f'(x)=[ln({3+x}/{3-x})]' (foloseste ln(u(x))'=1/u(x)*u(x)'

u(x)'=[{3+x}/{3-x}]'=[(3+x)'(3-x)-(3+x)(3-x)']/(3-x)^2=[3-x+3+x]/(3-x)^2=6/(3-x)^2

=>f'(x)=(3-x)/(3+x)*6/(3-x)^2=6/(3-x)(3+x), Observam ca f'(x)>0 pt orice x din (-3,3) =>f este crescatore pe (-3,3)