Connecting f, f', and f''

Relative max or Relative min

If f has a point of inflection, f' has a

Point of Inflection

If f" changes from positive to negative or negative to positive, f has a

Concave down.

If f" is negative, then f is

Decreasing

If f" is negative, then f' is

Concave up.

If f" is positive, then f is

Increasing

If f" is positive, then f' is

Potential Point of Inflection at x = a

If f" is zero or undefined at x = a, then f has a

Potential Relative Max or Min at x = a.

If f" is zero or undefined at x = a, then f' has a

Relative minimum

If f' changes from negative to positive, then f has a

Relative maximum

If f' changes from positive to negative, then f has a

Down

If f' is decreasing, f is concave ______

Up

If f' is increasing, f is concave ______

Decreasing

If f' is negative, then f is

Increasing

If f' is positive, then f is

Potential Relative Max or Min

If f' is zero or undefined at x = a, then f(x) has a _____________________________________________ at x = a.

Max

If x = a is a critical number (f'(a) = or or f'(a) is undefined) and f"(a) is negative, f has a relative _________ at x = a.

Min

If x = a is a critical number (f'(a) = or or f'(a) is undefined) and f"(a) is positive, f has a relative _________ at x = a.