Finding the Point(s) Where a Function Equals Its Average Value on an Interval

In this article, we will explore the process of finding the point(s) where a given function, f(x) = 2 - x^2, equals its average value on the interval [-6, 3].

Step 1: Calculate the average value of the function on the interval

To find the average value of a function f(x) on an interval [a, b], we use the formula:

Average value = (∫ f(x) dx from a to b) / (b - a)

In our case, the interval is [-6, 3], so we have:

Average value = (∫ (2 - x^2) dx from -6 to 3) / (3 - (-6)) Average value = (∫ 2 dx - ∫ x^2 dx from -6 to 3) / 9 Average value = (2(3 - (-6)) - (3^3 - (-6)^3) / 3) / 9 Average value = (2(9) - (27 - 216)) / 9 Average value = (18 - 189) / 9 Average value = -19 / 9

Step 2: Find the point(s) where the function equals its average value

To find the point(s) where the function f(x) = 2 - x^2 equals its average value, we need to solve the equation:

2 - x^2 = -19 / 9

Rearranging the equation, we get:

x^2 - 2 + 19 / 9 = 0

Solving this quadratic equation, we get:

x = ±√(2 - 19 / 9) x = ±√(17 / 9) x = ±√1.889 x = ±1.375

Therefore, the point(s) where the function f(x) = 2 - x^2 equals its average value on the interval [-6, 3] are:

x = 1.375 and x = -1.375

Conclusion

In this article, we have learned how to find the point(s) where a given function equals its average value on a specified interval. By following the step-by-step process, we were able to determine that the function f(x) = 2 - x^2 equals its average value of -19/9 at the points x = 1.375 and x = -1.375 on the interval [-6, 3].