we have the expression
[tex]g(x)=\frac{x^2+1}{f(x)}[/tex]
Find out the derivative g'(x)
so
[tex]g^{\prime}(x)=\frac{(2x)(f(x)-(x^2+1)\cdot f^{\prime}(x)}{(f(x))^2}[/tex]
Looking at the graph
f(2)=3
Find out the value of f'(x) at x=2
Find out the slope of f(x) between interval (0,3)
we have the points (0,-5) and (3,7)
m=(7+5)/(3-0)
m=12/3
m=4
so
f'(2)=4
substitute the given values in the expression above
[tex]g^{\prime}(2)=\frac{(2\cdot2)(3)-(2^2+1)\cdot4}{(3)^2}[/tex][tex]g^{\prime}(2)=\frac{(4)(3)-(5)\cdot4}{9}[/tex][tex]g^{\prime}(2)=-\frac{8}{9}[/tex]
therefore
the answer is option A
