Connecting features of f, f', f''
critical point
an x-intercept on f'(x) corresponds to a ___ of f(x).
f"(x)
concave up/down or positive to negative
local extrema
f'(x) changes sign The local maximum and minimum values of a function.
local minimum
f'(x) goes from negative to positive
local maximum
f'(x) goes from positive to negative
critical value
f'(x) value is zero (or undefined)
positive to negative
f(x) has a relative maximum when f'(x) changes from ___ to ___.
inflection points
f(x) has an ___ when f''(x) changes signs, or when f'(x) has a relative minimum or relative maximum.
f'(x)
is slope increasing/decreasing aka positive to negative
relative extrema
points of inflection on f(x) are ___ on the graph of f'(x).
y-values
the ___ of f'(x) give the slopes of f(x).
negative
when f''(x) is ___, f(x) is concave down.
relative extrema
when f'(x) changes signs, f(x) has ___.
concave up and positive
when f'(x) is increasing, f(x) is ___ and f"(x) is ___.
increasing
when f'(x) is positive, f(x) is ___.
slopes
when the ___ of f(x) are decreasing, f"(x) is negative.
f(x)
y value