Analyze Function Behaviors with Derivatives

Analyzing function's behaviors is one of the Core Purposes of studying Calculus.

AND THE CORE PURPOSE OF ANALYZING FUNCTION, IS FOR COMPUTER TO UNDERSTAND IT "BLINDLY", OR SAY "ALGEBRAICALLY"! BECAUSE IT CAN'T BE LIKE HUMAN TO "EYE BALL" IT!

First Derivative

• Function has critical points when f'(x) = 0

• Function has relative extrema when f'(x) crosses X-axis:

  • It has relative maxima when f'(x) crosses UP.

  • It has relative minima when f'(x) crosses DOWN.

Second Derivative

• Function has inflection points when f''(x)=0 and f''(x) crosses X-axis.

• It's concave up if f''(x) > 0

• It's concave down if f''(x) < 0

1st & 2nd Derivative

• It has a relative maximum when f'(x)=0 & f''(x) < 0

• It has a relative minimum when f'(x)=0 & f''(x) > 0

• It has an inflection point when f'(x) changes direction, OR f''(x) changes sign.

• It has a possible inflection point when f'(x) = 0 & f''(x) = 0.