Is #f(x) =(5x-2)/(2x+1)# concave or convex at #x=2#?

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Answer 1 · 2018-06-28T09:46:53

The function is concave

The function is #f(x)=(5x-2)/(2x+1)#

Calculate the first and second derivatives

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

#u=5x-2#, #=>#, #u'=5#

#v=2x+1#, #=>#, #v'=2#

#f'(x)=((5(2x+1)-2(5x-2)))/(2x+1)^2#

#=(10x+5-10x+4)/(2x+1)^2#

#=9/(2x+1)^2#

The second derivative is

#f''(x)=-9*2/(2x+1)^3*2#

#=-36/(2x+1)^3#

Therefore

When #x=2#

#f''(2)=-0.288#

As #f''(2)<0#, the function is concave

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