# How do you find the smallest and largest points of inflection for f(x)=1/(5x^2+3)?

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