Find the points at which y = f(x) = x^11-6x has a global maximum and minimum on the interval 0 ¢®A x ¢®A 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 can use the first derivative test and the critical points.
Step 1: Find the critical points.
A critical point is a point where the derivative of the function equals zero or is undefined. In this case, we need to find where f'(x) = 0.
Let's find the first derivative of f(x) = x^11 - 6x:
f'(x) = 11x^10 - 6.
Now, set f'(x) = 0 and solve for x:
11x^10 - 6 = 0.
Solving this equation, we find:
x^10 = 6/11
x ≈ 0.96003
So, the critical point is x ≈ 0.96003.
Step 2: Check the endpoints of the interval.
Since the interval is 0 ≤ x ≤ 4, we need to check the function values at x = 0 and x = 4.
When x = 0:
f(0) = 0^11 - 6(0) = 0.
When x = 4:
f(4) = 4^11 - 6(4) = 4194300.
Step 3: Determine the global maximum and minimum.
To determine if the critical point or the endpoints give the global maximum or minimum, we can compare the function values.
f(0) = 0
f(4) = 4194300
Since f(4) = 4194300 > 0 = f(0), the global maximum is at the point (4, 4194300).
Now let's find the function value at the critical point x ≈ 0.96003.
f(0.96003) ≈ 0.96003^11 - 6(0.96003) ≈ -4.14958
Since -4.14958 < f(0) = 0, the global minimum is at the point (0.96003, -4.14958).
So, the answers are:
Global Maximum: (x, y) ≈ (4, 4194300)
Global Minimum: (x, y) ≈ (0.96, -4.15)