+77 What, if anything, does the Mean Value Theorem gaurantee for the given function on the given interval?
f(x)=x2-2x+5 on [1,4]
+23246 f(x) = x2 - 2x - 5
When x = 1: f(1)) = 12 - 2(1) - 5 = 1 - 2 - 5 = -6 ---> one endpoint is (1, -6).
When x = 4: f(4)) = 42 - 2(4) - 5 = 16 - 8 - 5 = 3 ---> one endpoint is (4, 3).
The line through the endpoints (1, -6) and (4, 3) has slope: m = (3 - -6) / (4 - 1) = 9/3 = 3.
The Mean Value Theorem states that there is at least one point on the function f(x) = x2 - 2x - 5 in the open interval (1, 4) that has the same slope (m = 3).
