RELATIVE & ABSOLUTE EXTREMA

IDENTIFYING EXTREMA ON A GRAPH local or relative maximum local or relative minimum

IDENTIFYING EXTREMA ON A GRAPH The vertex of the parabola would be considered both a relative minumum and an absolute minimum

FUNCTIONS WITH RESTRICTED DOMAINS

HOW CAN WE FIND THESE HIGH AND LOW POINTS ON A GRAPH MATHEMATICALLY?

WHAT ARE “CRITICAL VALUES”? A CRITICAL VALUE is any value of 'x' where f'(x) = 0 or where f'(x) is undefined. MAXIMUMS AND MINIMUMS CAN ONLY OCCUR AT 1) CLOSED ENDPOINTS or 2) AT CRITICAL VALUES.

FINDING CRITICAL VALUES FIND THE CRITICAL VALUES FOR EACH FUNCTION BELOW. THEN, USE A GRAPHING UTILITY TO DETERMINE IF THIS CRITICAL VALUE WILL RESULT IN A RELATIVE MAXIMUM, RELATIVE MINIMUM OR NEITHER Example 1: Pierre de Fermat Prediction: relative minimum

Let’s look at the graph of the derivative

FOR FINDING RELATIVE EXTREMA THE 1ST DERIVATIVE TEST

The 1st Derivative Test Let x = c be a critical value for a function f(x) where f(c) exists. Then... If f' changes from + to - at x = c, then f has a relative maximum at x = c b) If f' changes from - to + at x = c, then f has a relative minimum at x = c

Let’s look at the graph of the derivative y’ < 0 to the left of x = 1.5 And y’ > 0 to the right of x = 1.5 So, by the 1st derivative test there must exist a relative minimum at x = 1.5

FINDING RELATIVE EXTREMA – NO CALCULATOR Example 2: Find all critical values and then determine if each critical value results in a maximum or minimum. Example 3: Find all critical values and then determine if each critical value results in a maximum or minimum. no maximums or minimums y’ > 0 to the left of x = -2 and y’ < 0 to the right of x = -2 So, by the 1st D.T. there must exist a relative maximum at x = -2 y’ < 0 to the left of x = 4/3 and y’ > 0 to the right of x = 4/3 So, by the 1st D.T. there must exist a relative minimum at x = 4/3

FINDING RELATIVE EXTREMA – NO CALCULATOR Example 4: Find all critical values and then determine if each critical value results in a maximum or minimum. y’ > 0 to the left and y’ < 0 to the right of x = /2 So, by the 1st D.T. there must exist a relative maximum at x = /2 y’ < 0 to the left and y’ > 0 to the right of x = 3/2 So, by the 1st D.T. there must exist a relative minimum at x = 3/2

Example 5: The function below represents the derivative of a function f that has a domain of all real numbers. Use f’ to determine the values of ‘x’ where function f has relative extrema. y’ > 0 to the left and y’ < 0 to the right of x = .2301 So, by the 1st D.T. there must exist a relative maximum at x = .2301 y’ < 0 to the left and y’ > 0 to the right of x = -2.897 So, by the 1st D.T. there must exist a relative minimum at x = -2.897

FINDING RELATIVE EXTREMA – CALCULATOR Example 6: Find all critical values and then determine if each critical value results in a maximum or minimum y’ < 0 to the left and y’ > 0 to the right of x = 0 So, by the 1st D.T. there must exist a relative minimum at x = 0

FINDING RELATIVE EXTREMA –CALCULATOR Example 7:Find all critical values and then determine if each critical value results in a maximum or minimum y’ > 0 to the left and y’ < 0 to the right of x = 0 So, by the 1st D.T. there must exist a relative maximum at x = 0 y’ > 0 to the right of x = -2 and x = -2 is a left endpoint So, there must exist a relative minimum at x = -2 y’ < 0 to the left of x = 2 and x = 2 is a right endpoint So, there must exist a relative maximum at x = 2

INCREASING & DECREASING

WHAT DO YOU REMEMBER ABOUT INCREASING & DECREASING FROM PRE-CALCULUS?

CALCULUS DEFINITIONS: Increasing: If 'f' is increasing on an interval of its domain then it is true that f'(x) ≥ 0 for all x on that interval. Decreasing: If 'f' is decreasing on an interval of its domain then it is true that f'(x)  0 for all x on that interval. Question: At what point(s) would a function change from increasing to decreasing?

INCREASING/DECREASING Example 1 – non-calculator: Use the derivative to find the exact intervals of increasing and decreasing. Increasing: Decreasing:

INCREASING/DECREASING Example 2 non-calculator: Use the derivative to find the exact intervals of increasing and decreasing. Increasing: Decreasing: none

INCREASING/DECREASING Example 3 non-calculator: Use the derivative to find the exact intervals of increasing and decreasing. Increasing: Decreasing:

INCREASING/DECREASING Example 4 non-calculator: Graph the following using a graphing utility. Predict the intervals of increasing and decreasing. Then use the derivative to find the exact intervals of increasing and decreasing. Decreasing:

Putting it all Together non-calculator Example 1: Find all relative maximum and minimums for the function below and find all intervals of increasing and decreasing.

Putting it all Together non-calculator Example 2: Find all relative maximum and minimums for the function below. Determine if any of these are absolute maximums or minimums.

Putting it all Together - calculator Example 3: Find all relative maximum and minimums for the function below and find all intervals of increasing and decreasing.