Analyzing f(x), f'(x), and f"(x)

yeah remember this? i dont.

f’(x) > 0

f(x) is increasing

f’(x) < 0

f(x) is decreasing

f’(x) is changing from - to +

f(x) has a local minimum

f’(x) is changing from + to -

f(x) has a local maximum

f’(x) is increasing

f(x) is concave up

f’(x) is decreasing

f(x) is concave down

f’(x) has an extreme value

f(x) has a point of inflection

(x,0)… f’(x)

if given f(x), the x coordinates of where max/min are occuring will be (__) on the _____ graph. (points on it’s x axis)

(x,0)

x-coordinates of points of inflection will be ____ on the **f”(x)** graph (points on the x-axis of f”)

when it looks like a U and has positive values on the f”(x) graph

given f(x), when is the function concave up?

when it looks like an upside down U and when f”(x) is negative

given f(x), when is the function concave down? (In terms of f”(x))

the x-coordinate will touch the x axis but not go through on the graph of f’(x)

given f(x), when there is a point on the function where the slope is zero but is NOT a max/min, _____________________.

when f(x) changes from concave up to concave down (or vice versa)

given f(x), when do points of inflection occur?

max/min…. sign change…. max/min

given f’(x), check the x-intercepts because they may be where the _____ of f(x) occur. there must be a _______ ______ for it to be a _____

positive

given f’(x), look for intervals where all of the y-values are ________. f(x) will be increasing on these intervals

negative

given f’(x), look for intervals where all of the y-values are _______. f(x) will be decreasing on these intervals

positive

given f’(x), intervals where f’(x) is increasing will show _________ y-values on the f”(x) graph

negative

given f’(x), intervals where f’(x) is decreasing will show ________ y-values on the f”(x) graph

concave up

given f’(x), intervals where f’(x) is incresasing means f(x) is _____________

concave down

given' f’(x), intervals where f’(x) is decreasing means f(x) is ______________

points of inflection…. points of inflection

if given f”(x), check x-intercepts. the x coordinates are the x coordinates of the _____________. if there are no x-intercepts, then there are no ______________.

positive

if given f”(x), look for intervals where all the y-values are _________. f’(x) will be increasing on those intervals

negative

if given f”(x), look for intervals where all the y-values are __________. f’(x) will be decreasing on those intervals.

increasing

if f’(x) and f”(x) have all positive values, then f(x) is _________. (think exponential functions)

decreasing

if f’(x) and f”(x) have all negative values, then f(x) is __________. (think negative exponential functions)

closed intervals

when answering a question along the lines of “on what intervals is f increasing/decreasing?” what kinds of intervals are you using?

open intervals

when answering a question along the lines of “on what intervals is f concave up/down?” what kind of interval are you using?