# What are the critical points of f(x) = x^2 + 2/x?

Dec 2, 2016

$\left(1 , 3\right)$ is the only critical point.

#### Explanation:

The critical points will occur when the derivative equals $0$ or is undefined .

$f ' \left(x\right) = 2 x + \frac{0 \times x - 1 \times 2}{x} ^ 2$

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

The derivative will be undefined at $x = 0$. However, the function is also undefined at $x = 0$, so this is not a critical point.

$0 = 2 x - \frac{2}{x} ^ 2$

$0 = \frac{2 {x}^{3} - 2}{x} ^ 2$

$0 = 2 {x}^{3} - 2$

$2 = 2 {x}^{3}$

$1 = {x}^{3}$

$x = 1$

The corresponding y-coordinate is

$f \left(1\right) = {1}^{2} + \frac{2}{1} = 3$

Hence, the only critical point is at $\left(1 , 3\right)$.

Hopefully this helps!