#"to determine if a function is concave/convex at x = a"#
#"we require to evaluate "f''(a)#
#• " if "f''(a)>0" then convex at x = a"#
#• " if "f''(a)<0" then concave at x = a"#
#f(x)=(x-3)^3/(x^2-9)#
#color(white)(f(x))=(x-3)^3/((x-3)(x+3))=((x-3)^2)/(x+3)#
#"differentiate using the "color(blue)"quotient rule"#
#"given "f(x)=(g(x))/(h(x))" then "#
#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larr" quotient rule"#
#g(x)=(x-3)^2rArrg'(x)=2(x-3)#
#h(x)=x+3rArrh'(x)=1#
#rArrf'(x)=((x+3)2(x-3)-(x-3)^2)/(x+3)^2#
#color(white)(rArrf'(x))=(x^2+6x-27)/(x+3)^2#
#"differentiate "f'(x)" using the "color(blue)"quotient rule"#
#g(x)=x^2+6x-27rArrg'(x)=2x+6#
#h(x)=(x+3)^2rArrh'(x)=2(x+3)#
#rArrf''(x)=((x+3)^2(2x+6)-(x^2+6x-27)2(x+3))/(x+3)^4#
#rArrf''(3)=(36(12)-0)/(1296)>0#
#rArrf(x)" is convex at "x=3#
graph{(x-3)^2/(x+3) [-10, 10, -5, 5]}
