The smallest point of inflection is at #x=-\frac{1}{\sqrt{5}}# and the largest point of inflection is as #x=\frac{1}{\sqrt{5}}#.
By the Quotient Rule, the first derivative is #f'(x)=\frac{-10x}{(5x^{2}+3)^{2}}# and the second derivative is #f''(x)=\frac{-10(5x^{2}+3)^{2}+10x\cdot 2(5x^{2}+3)\cdot 10x}{(5x^{2}+3)^{4}}# (the Chain Rule is also needed here).
The second derivative can be simplified:
#f''(x)=\frac{-50x^{2}-30+200x^{2}}{(5x^{2}+3)^{3}}=\frac{30(5x^{2}-1)}{(5x^{2}+3)^{3}}.#
Hence, #f''(x)=0# when #x=\pm\frac{1}{\sqrt{5}}#. Moreover, #f''(x)# changes sign at these values of #x#, making them #x#-coordinates of true inflection points.
