Relationship Between

f has a relative maximum where the graph of f ' crosses the x-axis from above to below.

f is decreasing where the graph of f ' is below the x-axis.

f has relative extrema where the graph of f ' crosses the x-axis.

f has inflection points where the graph of f ' has relative extrema.

f is concave up where the graph of f ' is increasing.

f has a critical number where the graph of f ' touches or crosses the x-axis.

f is increasing where the graph of f ' is above the x-axis.

f is concave down where the graph of f ' is decreasing.

f has a relative minimum where the graph of f ' crosses the x-axis from below to above.

If f ' is increasing, the graph of f is concave up.

If  f ' is positive, the graph of  f is increasing.

If  f ' is below the x-axis, the graph of f is decreasing.

If  f is concave down, the graph of  will be below the x-axis.

If  f ' is decreasing, the graph of f will be concave down.

If  f has an inflection point at , the graph of f ' has a relative extrema at .

If  f is concave up on (a,b), the graph of  will be above the x-axis on (a,b).

If  f is increasing on (a,b), the graph of  will be above the x-axis on (a,b).

If  f has a critical number at , the graph of  will touch or cross the x-axis at

Where the graph of  has relative extrema, f will have inflection points.

If the graph of is decreasing, the graph of will be below the x-axis.