On the interval ,the function y =
+ 3
- 9x + 27 has an absolute maximum when x =
A) 2.
B) 0.
C) 1.
E) none of the above

Question

On the interval  ,the function y = 
+ 3 
- 9x + 27 has an absolute maximum when x =
A) 2. 
B) 0. 
C) 1. 
E) none of the above

Quizplus · Accepted Answer

To find the absolute maximum of the function on the given interval, we need to find the critical points of the function and evaluate the function at those points as well as at the endpoints of the interval.

The derivative of the function is:
y' = 3(11ea83f6_6361_362a_b5a5_b30f255b427d_TB6578_11_TB6578_11) - 9
Setting y' = 0 to find critical points:
3(11ea83f6_6361_362a_b5a5_b30f255b427d_TB6578_11_TB6578_11) - 9 = 0
Simplifying:
11ea83f6_6361_362a_b5a5_b30f255b427d_TB6578_11_TB6578_11 = 3

Evaluating the function at the endpoints of the interval:
y(11ea83f6_6361_3628_b5a5_27ecf609b641_TB6578_11_TB6578_11) = 11ea83f6_6361_3629_b5a5_0fe084624611_TB6578_11_TB6578_11
+ 3(11ea83f6_6361_362a_b5a5_b30f255b427d_TB6578_11_TB6578_11)
- 9(11ea83f6_6361_3628_b5a5_27ecf609b641_TB6578_11_TB6578_11) + 27
Simplifying:
y(11ea83f6_6361_3628_b5a5_27ecf609b641_TB6578_11_TB6578_11) = 11ea83f6_6361_3629_b5a5_0fe084624611_TB6578_11_TB6578_11
+ 3(11ea83f6_6361_362a_b5a5_b30f255b427d_TB6578_11_TB6578_11)
- 9(11ea83f6_6361_3628_b5a5_27ecf609b641_TB6578_11_TB6578_11) + 27
= 11ea83f6_6361_3629_b5a5_0fe084624611_TB6578_11_TB6578_11 - 27

y(11ea83f6_6361_362b_b5a5_954933b7619c_TB6578_11_TB6578_11) = 11ea83f6_6361_3629_b5a5_0fe084624611_TB6578_11_TB6578_11
+ 3(11ea83f6_6361_362a_b5a5_b30f255b427d_TB6578_11_TB6578_11)
- 9(11ea83f6_6361_362b_b5a5_954933b7619c_TB6578_11_TB6578_11) + 27
Simplifying:
y(11ea83f6_6361_362b_b5a5_954933b7619c_TB6578_11_TB6578_11) = 11ea83f6_6361_3629_b5a5_0fe084624611_TB6578_11_TB6578_11
+ 3(11ea83f6_6361_362a_b5a5_b30f255b427d_TB6578_11_TB6578_11)
- 9(11ea83f6_6361_362b_b5a5_954933b7619c_TB6578_11_TB6578_11) + 27
= 11ea83f6_6361_3629_b5a5_0fe084624611_TB6578_11_TB6578_11 - 9(11ea83f6_6361_362b_b5a5_954933b7619c_TB6578_11_TB6578_11) + 30

From the above calculations, we can see that the function is decreasing on the interval (11ea83f6_6361_3628_b5a5_27ecf609b641_TB6578_11_TB6578_11, 11ea83f6_6361_362b_b5a5_954933b7619c_TB6578_11_TB6578_11) and increasing on the interval (11ea83f6_6361_362b_b5a5_954933b7619c_TB6578_11_TB6578_11, 11ea83f6_6361_3628_b5a5_27ecf609b641_TB6578_11_TB6578_11_TB6578_11). Therefore, the critical point we found is a local minimum. The function has no local maximum on the interval. The absolute maximum of the function on the interval is at the endpoint x = 2 since that point gives the largest value for y. Hence, the answer is A) 2.

On the Interval ,The Function Y =