# What are the points of inflection of f(x)=1/(5x^2+3x) ?

May 29, 2018

No Points of inflection

#### Explanation:

Writing
$f \left(x\right) = {\left(5 {x}^{2} + 3 x\right)}^{- 1}$
So
$f ' \left(x\right) = - {\left(5 {x}^{2} + 3 x\right)}^{- 2} \left(10 x + 3\right)$
$f ' \left(x\right) = - \frac{10 x + 3}{5 {x}^{2} + 3 x} ^ 2$
Using the Quotient rule to get

$f ' ' \left(x\right) = - \frac{10 {\left(5 {x}^{2} + 3 x\right)}^{2} - \left(10 x + 3\right) \cdot 2 \cdot \left(5 {x}^{2} + 3 x\right) \cdot \left(10 x + 3\right)}{5 {x}^{2} + 3 x} ^ 4$

Simplifying we get

$f ' ' \left(x\right) = \frac{6 \left(25 {x}^{2} + 15 x + 3\right)}{{x}^{2} \cdot {\left(5 x + 3\right)}^{2}}$