Curve Sketching/Concavity and Graphs

What are critical points?

Points where f′(x)=0 or f′(x) is undefined — possible extrema or flat points.

How do you determine if a function is increasing?

f′(x)>0: increasing

How do you determine if a function is decreasing?

f′(x)<0: decreasing

How do you find relative maxima?

• Use the First Derivative Test (check sign change of f′(x)

• Or the Second Derivative Test:

• f′(x)=0 and…

• f′′(x)<0 → max

How do you find relative minima?

• Use the First Derivative Test (check sign change of f′(x)

• Or the Second Derivative Test:

• f′(x)=0 and…

• f′′(x)>0 → min

What does concavity tell you?

• f′′(x)>0: concave up (cup)

• f′′(x)<0: concave down (cap)

What are inflection points?

Where f′′(x)=0 or undefined

• AND concavity changes sign

How do you find absolute extrema on a closed interval?

Evaluate f(x) at:

• All critical points within the interval

• The endpoints
  Compare all values to find the highest (max) and lowest (min)

How can you sketch f(x) from the graph of f′(x)?

• Where f′(x)>0: f increasing

• Where f′(x)<0: f decreasing

• Zeros of f′(x): possible max/min

How can you sketch f(x) from the graph of f′′(x)?

• f′′(x)>0: f is concave up

• f′′(x)<0: f is concave down

• Zeros of f′′(x): possible inflection points

What must you check to confirm an inflection point?

f′′(x)=0 and sign of f′′(x) changes around that point

What AP question types test curve sketching skills?

• "Where is f increasing/decreasing?"

• "Where are relative extrema?"

• "Where is f concave up/down?"

• "Where are inflection points?"

• "Match f, f′, or f′′ graphs"