1st, 2nd derivitive

f is increasing

f' > 0; f is moving upward

f is decreasing

f' < 0; f is moving downward

f has a horizontal tangent

f' = 0

f has a local maximum

f' changes from + to -; f' crosses the x-axis from above to below

f has a local minimum

f' changes from - to +; f' crosses the x-axis from below to above

f is concave up

f'' > 0; f' is increasing

f is concave down

f'' < 0; f' is decreasing

f has an inflection point

f'' changes sign; f' changes from increasing to decreasing or decreasing to increasing

f is increasing and concave up

f' > 0 and f'' > 0; f' is positive and increasing

f is increasing and concave down

f' > 0 and f'' < 0; f' is positive and decreasing

f is decreasing and concave up

f' < 0 and f'' > 0; f' is negative but increasing

f is decreasing and concave down

f' < 0 and f'' < 0; f' is negative and decreasing

f has a local maximum and is concave up

f' = 0 and changes + to -; f' is increasing on the relevant side

f has a local minimum and is concave up

f' = 0 and changes - to +; f' is increasing

f has a local maximum and is concave down

f' = 0 and changes + to -; f' is decreasing

f has a local minimum and is concave down

f' = 0 and changes - to +; f' is decreasing

f' is increasing

f'' > 0; f is concave up

f' is decreasing

f'' < 0; f is concave down

f' has a local maximum

f' changes from increasing to decreasing; f changes concavity from up to down

f' has a local minimum

f' changes from decreasing to increasing; f changes concavity from down to up

f'' > 0

f' is increasing; f is concave up

f'' < 0

f' is decreasing; f is concave down

f'' changes from positive to negative

f' changes from increasing to decreasing; f changes from concave up to concave down

f'' changes from negative to positive

f' changes from decreasing to increasing; f changes from concave down to concave up

What does f'(x) represent?

The slope of f(x)

What does f''(x) represent?

The slope of f'(x) and the concavity of f(x)