Which Points on the Graph Match These Calculus Conditions?

[Blade]

http://www.delteria.net/images/calctest.gif

Conditions:
a. f'(x)=0
b. f"(x)= 0
c. f'(x) = DNE
d. f = relative max
e. f = point of inflection

What I have so far (they can repeat I believe):
a. X0, X4
b. X3
c. X1
d. X2
e. X2

I'm sure something is wrong... Also, what would a f"=0 look like?

 

[HallsofIvy]

This really should be under "homework".

a) Yes, at x0 and x4, the tangent line is horzontal so f'= 0.

b) f"= 0 means the curve has 0 "curvature"(!) and so is very "straight" at least for a short distance. I would agree that it looks like the curve is very straight at x3 but I recommend you also look closely at x2. f"> 0 means the curve is "concave" upward while f"< 0 means it is concave downward. f"= 0 where the concavity changes.

c) Yes, there is a "cusp" at x1 and so the derivative does not exist.

d) "relative max" should be easiest of all but it surely doesn't happen at x2! Forget about derivatives and just ask yourself "where does the curve to up to the point and then back down again?"

e) A "point of inflection" is where the second derivative exists but changes sign (and so must be 0). Look at (b) again.

 

[M98Ranger]

Your response is mostly correct. Here are a few corrections and clarifications:

a. f'(x)=0: This condition means that the slope of the function at that point is 0. In the provided graph, this occurs at X0 and X4. So your response is correct.

b. f"(x)=0: This condition means that the second derivative of the function is 0. In the provided graph, this occurs at X3. So your response is correct.

c. f'(x) = DNE: This condition means that the derivative of the function does not exist at that point. In the provided graph, this occurs at X1. So your response is correct.

d. f = relative max: This condition means that the function has a maximum value at that point. In the provided graph, this occurs at X2. So your response is correct.

e. f = point of inflection: This condition means that the concavity of the function changes at that point. In the provided graph, this occurs at X2. So your response is correct.

For f''=0, it would look similar to b. f"(x)=0, except the point would be labeled as X2 instead of X3. This is because the second derivative is 0 at X2, indicating that the concavity changes at that point.