Concaves Flashcards
If f ‘ (x) > 0, then…
f(x) is increasing
If f ‘ (x) < 0, then…
f(x) is decreasing
If f ‘ (x) = 0, then…
f(x) has a horizontal tangent
f ‘’ (x) > 0 means that f(x)…
It concaves up (like a cup)
f ‘’ (x) < 0 means that f(x)…
Concaves down (like a frown)
A point of inflection is a point on the curve where…
Concavity changes
To find a point of inflection…
Look for where f ‘’ changes signs, or equivalently, where f ‘ changes direction
To find extreme values of a function, look for where…
f ‘ is zero or undefined (critical numbers)
At a maximum, the value of the derivative…
f ‘ changes from positive to negative (First Derivative Test)
At a minimum, the value of the derivative…
f ‘ changes from negative to positive (First Derivative Test)
The Second Derivative Test:
If f ‘ (x) = 0 and…
f ‘’ (x) < 0, then f has a maximum; if f ‘’ (x) > 0, then f has a minimum
