Question

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]

Answer 1

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

The point "c" of interest is c = {5}