# What are the local extrema, if any, of f (x) =(x^3+3x^2)/(x^2-5x)?

Jun 3, 2018

$M I N \left(5 + 2 \sqrt{10} , 4 \sqrt{10} + 13\right)$ and $M A X \left(5 - 2 \cdot \sqrt{10} , - 4 \sqrt{10} + 13\right)$

#### Explanation:

By the Quotient rule we get

f'(x)=((3x^2+6x)(x^2-5x)-(x^3+3x^2)(2x-5))/(x^2-5x)^2
simplifying we obtain

$f ' \left(x\right) = \frac{{x}^{2} - 10 x - 15}{x - 5} ^ 2$

$f ' ' \left(x\right) = \frac{80}{x - 5} ^ 3$

so we have

$f ' ' \left(5 + 2 \cdot \sqrt{10}\right) = \frac{80}{2 \sqrt{10}} ^ 3 > 0$

$f ' ' \left(5 - 2 \sqrt{10}\right) = \frac{80}{- 2 \sqrt{10}} ^ 3 < 0$