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

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xalal | (Level 1) Honors

Posted June 27, 2013 at 5:39 PM via web

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

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted June 27, 2013 at 5:49 PM (Answer #1)

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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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