Please help me sketch this graph:
f ' (x) > 0 when x<2 and when 2< x <5
f ' (x) < 0 when x > 5
f ' (2) = 0
f " (x) < 0 when x < 2 and 4 < x < 7
f " (x) > 0 when 2 < x < 4 and x > 7
Just one piece at a time.
f ' (x) > 0 when x < 2
Up from the left.
f ' (2) = 0
Level off at x = 2
f ' (x) > 0 when 2 < x < 5
Surprise! Don't trun around and start down at x = 2. (Think about $\displaystyle y\;=\;x^{3}$ at the Origin)
f ' (x) < 0 when x > 5
Okay, NOW turn around and start down. Notice, however, that there must be a point of inflection between x=2 and x=5.
f " (x) < 0 when x < 2 and 4 < x < 7
f " (x) > 0 when 2 < x < 4 and x > 7
Aw! They gave away the point of inflection at x = 4. Further, they gave us another one at x = 7, so don't be wandering off the the bottom of the world.