f =ln (1+2/x) is concave if x<-2?explain

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Given `f=ln(1+2/x)` :

The domain of the function is `(-oo,-2)uu(0,oo)` . We are interested in the interval `(-oo,-2)` .

`f'(x)=(-2/x^2)/(1+2/x)=(-2)/(x^2+2x)`

`f''(x)=((x^2+2x)(0)-(-2)(2x+2))/((x^2+2x)^2)=(4(x+1))/((x^2+2x)^2)`

Note that the denominator is always positive. On the interval `(-oo,-2)` the numerator is always negative.

**Thus the function is concave down for x<-2.**

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