Answer:
Average value A = 2
c = 0.5 or c = -0.5
Step-by-step explanation:
The average value A of a function f(x) between the points (a,b) is given by
[tex]A = \frac{1}{b-a}\int\limits^b_a {f(x)} \, dx [/tex]
You have [tex]f(x) = |4x|[/tex].
From the definition of the module equation, we have that:
[tex]\left \{ {{|f(x)|=f(x), f(x) \geq 0} \atop {|f(x)|= - f(x), f(x) < 0}} \right [/tex]
So
[tex]\left \{ {{|4x|=4x, x \geq 0} \atop {|4x|= - 4x, x < 0}} \right[/tex]
The average value will then be
[tex]A = \frac{1}{1-(-1)} \int\limits^0_{-1} {-4x} \, dx + \int\limits^1_{0} {4} \, dx [/tex]
[tex]A = \frac{1}{2} {\int\limits^0_{-1} {-4x} \, dx + \int\limits^1_{0} {4x} \, dx}[/tex]
[tex]A = \frac{4}{2} [/tex]
So the average value A = 2.
As for the second question
|4x| = 2 when
4x = 2 or 4x = -2
x = 0.5 or x = -0.5
So there are two values of c in this interval that are equal to the average value: c = 0.5 or c = -0.5
We will see that the average is equal to 0.
For a function f(x) we define the average in an interval [a, b] as:
[tex]\frac{f(b) - f(a)}{b - a}[/tex]
In this case we have:
f(x) = |4x|
And the interval is [-1, 1]
Replacing that we get:
[tex]\frac{|4*1| - |4*-1|}{1 - (-1)} = 0[/tex]
So the average of the function f(x) = |4x| on the given interval is 0.
If you want to learn more about averages, you can read:
https://brainly.com/question/8728504
