Questions: Find the point(s) at which the function (f(x)=7-2 x) equals its average value on the interval ([0,4]). The function equals its average value at (x=) (square) (Use a comma to separate answers as needed.)

Solution Steps

The average value of the function $f(x) = 7 - 2x$ on the interval $[0, 4]$ is calculated as follows: $$\text{Average value} = \frac{1}{{x_2 - x_1}} \int_{{x_1}}^{{x_2}} (a-bx) dx$$ $$= \frac1{4 - 0} \left( 7(4 - 0) - \frac{2}2(4^2 - 0^2) \right)$$ $$= 7 - \frac{2}2(4 + 0)$$ $$= 3$$

Setting the function equal to its average value and solving for $x$ gives: $$a - bx = 7 - \frac{2}2(4 + 0)$$ Simplifying and solving for $x$ yields: $$x = \frac12(4 + 0)$$ $$x = 2$$

Final Answer