Verify that the function satisfies the three hypotheses of rolle's theorem on the given interval. then find all numbers c that satisfy the conclusion of rolle's theorem. (enter your answers as a comma-separated list.) f(x) = 4 − 20x + 2x2, [4, 6]
f(x) is a polynomial with real coefficients. Hence for any real numbers x, ... • f(x) is a real-valued function • f(x) is continuous on any closed interval of real numbers, such as [4, 6] • f(x) is differentiable on any open interval of real numbers, such as (4, 6)
The conclusion is that there exists some "c" such that f'(c) = (f(6) -f(4)/(6 - 4).
The slope of interest is m = (f(6) -f(4))/(6 -4) = (-44 -(-44))/2 = 0
The slope f'(x) is -20 +4x. It will be zero where 0 = -20 +4x 20 = 4x 5 = x So, f'(5) = 0 = m
