To find the absolute extreme values (maximum or minimum) of a function on a given domain, we need to follow these steps:
1. Find the critical points of the function inside the domain.
2. Evaluate the function at the critical points and at the endpoints of the domain.
3. Compare the values obtained to determine the absolute extreme values.
Let's apply these steps to the function f(x) = 2x^2 + 384 ln(x) on the domain [1,6].
1. Find the critical points of the function inside the domain.
The critical points of a function are the points where the derivative is zero or undefined. We can find the derivative of f(x) as:
f'(x) = 4x + 384/x
To find the critical points, we need to solve the equation f'(x) = 0:
4x + 384/x = 0
Multiplying both sides by x, we get:
4x^2 + 384 = 0
Dividing both sides by 4, we get:
x^2 + 96 = 0
This equation has no real solutions, since x^2 is always positive and 96 is positive too. Therefore, there are no critical points inside the domain [1,6].
2. Evaluate the function at the critical points and at the endpoints of the domain.
Since there are no critical points inside the domain, we only need to evaluate the function at the endpoints:
f(1) = 2(1)^2 + 384 ln(1) = 0
f(6) = 2(6)^2 + 384 ln(6) ≈ 290.7
3. Compare the values obtained to determine the absolute extreme values.
The function f(x) has a minimum value of 0 at x = 1 and a maximum value of approximately 290.7 at x = 6.
Therefore, the absolute extreme values of the function f(x) on the domain [1,6] are:
- Absolute minimum: 0, at x = 1
- Absolute maximum: approximately 290.7, at x = 6