
Maximum and Minimum Values 
DEFINITION: A function has an absolute maximum (or global maximum) at
if for all in where is the domain of The number is called
the maximum value of on Similarly, has an absolute minimum (or global minimum) at
if for all in and the number is called
the minimum value of on The maximum and minimum values of are called the extreme values
of
DEFINITION: A function has a local maximum (or relative maximum) at if when is near [This means that for all in some open interval containing ] Similarly, has a local minimum (or relative minimum) at if when is near 1. Find the absolute and local maximum and minimum values of the function The function has the absolute (and local) minimum at and has no absolute or local maximum. 2. Find the absolute and local maximum and minimum values of the function The function has no absolute or local minimum and no absolute or local maximum. 3. Find the absolute and local maximum and minimum values of the functions etc.
The functions etc. have no absolute or local minima and no absolute or local maxima. 4. Find the absolute and local maximum and minimum values of the functions The functions have infinitely many absolute and local minima and maxima. 5. Find the absolute and local maximum and minimum values of the functions The functions have infinitely many local minima and maxima and no absolute minima and maxima. 6. Find the absolute and local maximum and minimum values of the functions The functions have no absolute or local minima and no absolute or local maxima. 7. Find the absolute and local maximum and minimum values of the function The function has the absolute minimum at and has no absolute maximum. It has two local minima at and and the local maximum at 8. Find the absolute and local maximum and minimum values of the function The function has the absolute minimum at and absolute maximum at It has no local maximum or minimum. 9. Find the absolute and local maximum and minimum values of the function The function has no absolute or local minimum and no absolute or local maximum. 10. Find the absolute and local maximum and minimum values of the function The function has no absolute or local minimum and no absolute or local maximum. 11. Find the absolute and local maximum and minimum values of the function The function has the absolute (and local) minimum and absolute (and local) maximum at any point on the number line. THEOREM (The Extreme Value Theorem): If is continuous on a closed interval then attains an absolute maximum value and an absolute minimum value at some numbers and in THEOREM (Fermat's Theorem): If has a local maximum or minimum at and if exists, then REMARK 1: The converse of this theorem is not true. In other words, when does not necessarily have a local maximum or minimum. For example, if then equals at but is not a point of a local minimum or maximum. REMARK 2: Sometimes does not exist, but is a point of a local maximum or minimum. For example, if then does not exist. But has its local (and absolute) minimum at DEFINITION: A critical number of a function is a number in the domain of such that either or does not exist. REMARK: From Fermat's Theorem it follows that if has a local maximum or minimum at then is a critical number of (a) Find the critical numbers of . Since the only critical number of is (b) Find the critical numbers of . Since (c) Find the critical numbers of . We have Since is not in the domain, has no critical numbers. (d) Find the critical numbers of We have thus at and Since exists everywhere, and are the only critical numbers. (e) Find the critical numbers of We have or In short, It follows that equals when which is at does not exist when which is at Thus the critical numbers are and (f) Find the critical numbers of We have In short, Here is an other way to get the same result: or In short, It follows that equals when which is at does not exist when which is at Thus the critical numbers are and THE CLOSED INTERVAL METHOD: To find the absolute maximum and minimum values of a continuous function on a closed interval : 1. Find the values of at the critical numbers of in 2. Find the values of at the endpoints of the interval. 3. The largest of the values from Step 1 and 2 is the absolute maximum value; the smallest value of these values is the absolute minimum value. EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: Since there are no critical numbers. Step 2: Since there are no critical numbers, we evaluate at the endpoints and only. We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: Since the number at which is zero is . Therefore is the critical number. Step 2: We evaluate at the critical number and at the endpoints We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: Since the critical number is Step 2: We evaluate at the critical number and at the endpoints We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: Since the critical numbers are and Step 2: We evaluate at the critical numbers and at the endpoints We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at and EXAMPLE: Find the absolute maximum and minimum values of on the interval and determine where these values occur. Step 1: Since there are two critical numbers and Step 2: We now evaluate at these critical numbers and at the endpoints and We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: Since there are two critical numbers and Step 2: Since is not from we evaluate only at and at the endpoints and We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: Since the critical numbers are and Step 2: Since We evaluate only at the critical numbers and at the endpoints . We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: We have Note that there are no numbers at which is zero. The number at which does not exist is , but this number is not from the domain of Therefore does not have critical numbers. Step 2: Since does not have critical numbers, we evaluate it only at the endpoints and We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: We have we have The numbers at which is zero are and The number at which does not exist is . But is not defined at , therefore the critical numbers of are and only. Step 2: Since we evaluate only at the critical number (which is also one of the endpoints) and at the second endpoint . We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: We have Note that there are no numbers at which is zero. The number at which does not exist is , so the critical number is Step 2: We evaluate at the critical number and at the endpoints and We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: We have Note that there are no numbers at which is zero. The number at which does not exist is , so the critical number is Step 2: We evaluate at the critical number and at the endpoints and We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: We have Note that there are no numbers at which does not exist. The number at which is zero is , so the critical number is Step 2: We evaluate at the critical number and at the endpoints and We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: We have Note that the number at which is zero is and the numbers at which does not exist are . Therefore the critical numbers are Step 2: Since we evaluate only at the critical numbers (which is also one of the endpoints) and at the second endpoint We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on the interval and determine where these values occur. Step 1: Since there are two critical numbers and Step 2: We now evaluate at these critical numbers and at the endpoints and We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at EXAMPLE: Find the absolute maximum and minimum values of on and determine where these values occur. Step 1: We have Note that there are no numbers at which does not exist. The numbers in at which is zero on are , so the critical number is Step 2: We evaluate at the critical numbers and at the endpoints We have Step 3: The largest value is and the smallest value is Therefore the absolute maximum of on is occurring at and the absolute minimum of on is occurring at 