Is #f(x) =(x-2)/(x+1)# concave or convex at #x=2#?

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Answer 1 · 2018-03-13T14:32:20

The function is concave.

Calculate the first and second derivative

#(u/v)=(u'v-uv')/(v^2)#

#u=(x-2)#, #=>#, #u'=1#

#v=(x+1)#, #=>#, #u'=1#

So,

#f'(x)=(1*(x+1)-(x-2)*1)/(x+1)^2=3/(x+1)^2#

#f''(x)=-6/(x+1)^3#

Next, calculate the sign of #f''(2)#

#f''(2)=-6/(27)=-2/9#

As,

#f''(x)<0#,

#f(2)# is concave at #x=2#

graph{(x-2)/(x+1) [-10, 10, -5, 5]}