# What are the critical values, if any, of f(x)=(x^2-5x)/(x^2-2)?

Nov 4, 2015

$f$ has no critical values.

#### Explanation:

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

Note that the domain of $f$ is all real numbers except $\pm \sqrt{2}$

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

$= \frac{2 {x}^{3} - 5 {x}^{2} - 4 x + 10 - 2 {x}^{3} + 10 {x}^{2}}{{x}^{2} - 2} ^ 2$

$= \frac{5 {x}^{2} - 4 x + 10}{{x}^{2} - 2} ^ 2$

$f ' \left(x\right)$ is not defined for $x = \pm \sqrt{2}$, but those numbers are not in the domain of $f$, so they are not critical values for $f$.

$f ' \left(x\right) = 0$ when $5 {x}^{2} - 4 x + 10 = 0$. The discriminant of this quadratic is negative, so there are no real zeros.

$f$ has no critical numbers.