Find the absolute maximum and absolute minimum values of the function
f(x)=(x−2)(x−5)^3+11 on each of the indicated intervals.
Enter -1000 for any absolute extrema that does not exist.
(A) Interval = [1,4]
Absolute maximum =
Absolute minimum =
(B) Interval = [1,8] .
Absolute maximum =
Absolute minimum =
(C) Interval = [4,9] .
Absolute maximum =
Absolute minimum =
My answers are:
(A) Interval = [1,4]
Absolute maximum = 9
Absolute minimum = -5
(B) Interval = [1,8] .
Absolute maximum = 173
Absolute minimum = -5
(C) Interval = [4,9] .
Absolute maximum = 459
Absolute minimum = -1000
Only the max for the second one and the max for the third one are right.... I don't know what to do....
To find the absolute maximum and minimum values of a function on an interval, you can follow these steps:
1. Take the derivative of the function with respect to x to find the critical points (where the derivative is zero or undefined) and determine where the function increases or decreases.
2. Evaluate the function at the critical points and the endpoints of the interval.
3. Compare the values obtained in step 2 to determine the absolute maximum and minimum values.
Let's work through each interval:
(A) Interval = [1,4]
1. Take the derivative of the function:
f'(x) = 4(x - 5)^2(x - 2) + (x - 2)(3(x - 5)^2)
Simplifying, we get:
f'(x) = (x - 2)(4(x - 5)^2 + 3(x - 5)^2)
f'(x) = (x - 2)(7(x - 5)^2)
The critical point is x = 2, since it makes the derivative zero.
2. Evaluate the function at the critical point and endpoints:
f(1) = (-1)(-4)^3 + 11 = -91
f(4) = (2)(-1)^3 + 11 = 9
f(2) = (0)(-3)^3 + 11 = 11
3. Compare the values:
The absolute maximum is 11 (at x = 2), and the absolute minimum is -91 (at x = 1).
(B) Interval = [1,8]
1. Take the derivative of the function:
Using the same derivative as before,
f'(x) = (x - 2)(7(x - 5)^2)
The critical point is still x = 2.
2. Evaluate the function at the critical point and endpoints:
f(1) = -91
f(8) = (6)(3)^3 + 11 = 173
f(2) = 11
3. Compare the values:
The absolute maximum is 173 (at x = 8), and the absolute minimum is -91 (at x = 1).
(C) Interval = [4,9]
1. Take the derivative of the function:
Using the same derivative as before,
f'(x) = (x - 2)(7(x - 5)^2)
There are no critical points within this interval.
2. Evaluate the function at the endpoints:
f(4) = 459
f(9) = (7)(4^2)(2^3) + 11 = 459
3. Compare the values:
The absolute maximum is 459 (at x = 4 and x = 9), and the absolute minimum is -1000 (as it does not exist within this interval).
Based on these steps, the correct answers are:
(A) Interval = [1,4]
Absolute maximum = 11
Absolute minimum = -91
(B) Interval = [1,8]
Absolute maximum = 173
Absolute minimum = -91
(C) Interval = [4,9]
Absolute maximum = 459
Absolute minimum = -1000