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