# What are the absolute extrema of  f(x)= 2 + x^2 in [-2, 3]?

Mar 5, 2017

$f \left(x\right)$ has an absolute minimum of 2 at $x = 0$

#### Explanation:

$f \left(x\right) = 2 + {x}^{2}$

$f \left(x\right)$ is a parabola with a single absolute minimum where $f ' \left(x\right) = 0$

$f ' \left(x\right) = 0 + 2 x = 0 \to x = 0$

$\therefore {f}_{\min} \left(x\right) = f \left(0\right) = 2$

This can be seen on the graph of $f \left(x\right)$ below:

graph{2+x^2 [-9.19, 8.59, -0.97, 7.926]}