Final answer:
To find the point(s) at which the function f(x) = 10 - x equals its average value on the interval, we need to find the average value of the function on the given interval first. The average value of a function on an interval is equal to the integral of the function over the interval divided by the length of the interval.
Step-by-step explanation:
In order to find the point(s) at which the function f(x) = 10 - x equals its average value on the interval, we need to find the average value of the function on the given interval first. The average value of a function on an interval is equal to the integral of the function over the interval divided by the length of the interval.
The interval in this case is 0 ≤ x ≤ 20. The length of the interval is 20 - 0 = 20.
The integral of the function f(x) = 10 - x over the interval 0 ≤ x ≤ 20 is:
∫ (10 - x) dx = 10x - (1/2)x^2
The average value of the function on the interval is:
A = (1/20) * ∫ (10 - x) dx = (1/20) * (10x - (1/2)x^2)
To find the point(s) at which the function equals its average value, we set f(x) = A and solve for x:
10 - x = (1/20) * (10x - (1/2)x^2)
Simplifying the equation:
20 - 20x = 10x - (1/2)x^2
Combining like terms and rearranging:
(1/2)x^2 - 30x + 20 = 0
Now we can solve this quadratic equation for x using factoring, completing the square, or the quadratic formula.