Finding Extreme Values: Closed vs. Open Intervals

[donjt81]

So I know how to find extreme value when it is a closed interval
for example

f(x) = x^2 - 1, -1 <= x <= 2
in this i would first find the critical point. and then i would compare f(critical point) and f(-1) and f(2) and then find the maximum and minimum values that way.

but my question is how to find extreme values when it is not a closed interval. Right now what I am doing is I plot the graph on my calculator and look at the max and the minimum values. But i am sure that's not the correct way of doing it. I'm sure there is some way to find the max and the min algebraically.

here is an example of a problem without closed intervals
f(x) = 2x^2 - 8x + 9

Thanks in advance

 

[donjt81]

anyone...?

 

[neutrino]

Have you tried the second derivative test?

 

[HallsofIvy]

A continuous function does not necessarily HAVE a maximum or minimum on an open interval. Start in exactly the way you did for a closed interval. Find the critical points, evaluate at those critical points and the end points. The difference is: if the value at one end point is larger than at any of the critical points, the function does NOT have a maximum in that interval. If the value at one end point is smaller than at any of the critical points, the function does NOT have a minimum in that interval.

(I'm assuming that the function is continuous at both end point.)