Polynomial functions

Question

Determine where the absolute maximum value and the absolute minimum value of the function

f(x) = x² - 5x + 3 occur over the specified interval:

 

a. [-2, 4]

b. [-2, 10]

c. [-2, 1]

d. [3, 4]

e. [3/2, 7,2]

Answer 1

To clarify on Kathy's answer:  

 

The reason she is evaluating the endpoints of the intervals is because if a function has a restricted domain, then an absolute extremum can occur on the endpoints of the interval of definition.  

 

For instance, the function f(x)=x defined on (-∞,∞) has no extrema whereas f(x)=x defined on [-1,1] has an absolute minimum at x=-1 and an absolute maximum at x=1.  (Think of the graph of the second function for further justification.)

Answer 2

The derivative of this function is 2x - 5.

Set 2x - 5 = 0 and you get x = 2/5.  This is a critical number and is in the first problem's interval.

 

Now evaluate x = 2/5 in the original function.

(2/5)^2 -5(2/5) + 3 = 29/25

 

Also, test the endpoints in the original function.

Evaluate x = -2 so (-2)^2 -5(-2)+3 = -3

Evaluate x = 4 so (4)^2-5(4)+3 = -1

 

Find the largest of these 3 answers and that is the absolute maximum in this first interval. ...29/25

 

Find the smallest of these three answers and that is the absolute minimum in the first interval....-3

 

Repeat for each interval.  The critical number is always x = 2/5  and cannot be considered for problems d and e because it is not in the given interval.  So just evaluate the endpoints.

Answer 3

To find the maximum or minimum points of the equation y = ax² + bx + c  we have to 

1. take derivative and set it equal to 0, we get: 2ax + b = 0
2. solve for "x", gives: x = (- b)/2a
3. if "a" is positive, then x = (- b)/2a is a minimum. If "a" is negative, x = (- b)/2a is a maximum. 
~~~~~~~~~~~~  

f ' (x) = 2x - 5

2x - 5 = 0 ----> x = 5/2 = 2.5

a > 0 ----> x = 2.5 is a minimum of a given function

f(2.5) = - 3.25

Thus, chose [(a), (b), (e)] is the answer.