Calculus graph element spotting

Blade

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.