Can a Point Be Both a Local Minimum and Maximum?

[chubbyorphan]

Hey forum I got to submit this in a few hours so if anyone could help me with this quick times, you would really be saving me :P

Homework Statement

Given the following graph of h(x)
I only need help with part b)
which asks for the local minimum or maximum points of the graph

http://i43.tinypic.com/hvyrk4.jpg

The Attempt at a Solution

So basically I realize there's a horizontal tangent at the point h(2)

What I can't decide is if there is neither a minimum nor maximum (my initial thoughts)


but then one part of my book reads for ANOTHER graph:
f'(x) is never zero, so the function has no local maximums or minimums

and we can clearly see that h(x) has a horizontal tangent implying that h'(x) would be zero at h'(2).. so
is the point [2,h(2)] both a local minimum and a local maximum

Is that even possible for a point to be both a local maximum and a local minimum?

if someone could clarify as to why its whichever answer that would be awesome!
argh please help!

 

[jbsteele]

No, it is not possible for a point to be both a local maximum and a local minimum. The point [2,h(2)] is neither a local minimum nor a local maximum. This is because, while the derivative of h(x) is zero at x=2, the graph does not have an inflection point there, which would be necessary for a point to be both a local maximum and minimum.