Is #f(x) =(x-3)/(3x+1)# concave or convex at #x=-1#?

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Answer 1 · 2018-07-27T20:53:11

function is convex at #x=-1#

Given function

#f(x)=\frac{x-3}{3x+1}#

#f'(x)=\frac{3x+1-(x-3)(3)}{(3x+1)^2}#

#f'(x)=\frac{10}{(3x+1)^2}#

#f''(x)=\frac{10(-2)(3)}{(3x+1)^3}#

#f''(x)=-\frac{60}{(3x+1)^3}#

#f''(-1)=-\frac{60}{(3(-1)+1)^3}#

#=7.5>0#

Since #f''(-1)>0# hence the given function is convex at #x=-1#