Find the points at which y = f(x) = x^11-6x has a global maximum and minimum on the interval 0 ¡ x ¡ 4 Round your answers to two decimal places.

Global Maximum:
(x,y) = (,)
Global Minimum:
(x,y) = (,)

To find the global maximum and minimum of the function y = f(x) = x^11-6x on the interval 0 ¡ x ¡ 4, we need to find the critical points first.

To find the critical points, we need to find the values of x where the derivative of the function is equal to zero or does not exist.

Step 1: Find the derivative of f(x):

f'(x) = 11x^10 - 6

Step 2: Set the derivative equal to zero and solve for x:

11x^10 - 6 = 0

Solving for x, we get:

x^10 = 6/11

x = (6/11)^(1/10) ≈ 0.968

Since x = 0.968 is within the interval 0 ¡ x ¡ 4, we can proceed to find the values of y at x = 0, x = 0.968, and x = 4 to determine the global maximum and minimum.

Step 3: Find the value of y at x = 0:

y = f(0) = 0^11 - 6(0) = 0

Step 4: Find the value of y at x = 0.968:

y = f(0.968) = (0.968)^11 - 6(0.968) ≈ -4.522

Step 5: Find the value of y at x = 4:

y = f(4) = 4^11 - 6(4) = 4194304 - 24 = 4194280

Therefore, the points at which the function has a global maximum and minimum on the interval 0 ¡ x ¡ 4 are:

Global Maximum: (x, y) ≈ (4, 4194280)
Global Minimum: (x, y) ≈ (0.968, -4.522)

To find the points at which the function y = x^11 - 6x has a global maximum and minimum on the interval 0 < x < 4, you will need to follow these steps:

1. Find the critical points of the function:
Critical points occur when the derivative of the function is equal to zero or undefined. So, let's find the derivative of the function:
f'(x) = 11x^10 - 6

Now, set f'(x) = 0 and solve for x:
11x^10 - 6 = 0

Solve for x:
11x^10 = 6
x^10 = 6/11

Take the 10th root of both sides to solve for x:
x = (6/11)^(1/10)

2. Check the endpoints of the interval:
Since the interval is 0 < x < 4, we need to check the function at both x = 0 and x = 4.

3. Evaluate the function at the critical points and endpoints:
Plug in the values of x into the function to find the corresponding y-values.
- Evaluate at x = (6/11)^(1/10)
- Evaluate at x = 0 (endpoint)
- Evaluate at x = 4 (endpoint)

4. Compare the y-values to find the global maximum and minimum:
Identify the largest and smallest y-values obtained from step 3.

1. Find the critical points:
By solving the equation 11x^10 - 6 = 0, we find that x = (6/11)^(1/10).

2. Check the endpoints:
Since the interval is 0 < x < 4, we need to evaluate the function at x = 0 and x = 4.

- f(0) = (0^11) - 6(0) = 0
- f(4) = (4^11) - 6(4) = 4194304 - 24 = 4194280

3. Evaluate the function at the critical point and endpoints:
- f((6/11)^(1/10))
Calculate the value of (6/11)^(1/10) using a calculator or software. Let's say it's approximately 0.49178.
Evaluating f(0.49178), we get:
f(0.49178) = (0.49178^11) - 6(0.49178) = 0.011640 - 2.950648 = -2.939008

- f(0) = 0 (already calculated in step 2)
- f(4) = 4194280 (already calculated in step 2)

4. Compare the y-values:
The y-values obtained are as follows:
- f((6/11)^(1/10)) ≈ -2.939008
- f(0) = 0
- f(4) = 4194280

The global maximum is the largest y-value, which is f(4) = 4194280 at x = 4.
The global minimum is the smallest y-value, which is f((6/11)^(1/10)) ≈ -2.939008 at x ≈ 0.49178.

Therefore, the points at which the function has a global maximum and minimum on the interval 0 < x < 4 are:

Global Maximum: (x, y) = (4, 4194280)
Global Minimum: (x, y) ≈ (0.49178, -2.939008)