Verify that Rolle's Theorem can be applied to the function f(x)=x3−7x2+14x−8 on the interval [1,4]. Then find all values of c in the interval such that f′(c)=0.

Answer 1

f(1) = 1 - 7 + 14 - 8 = 0; f(4) = 64 - 112 + 56 - 8 = 0. So, f(1) = f(4) and f(x) differentiable as polynomial. and Rol's Theorem working.

f'(x) = 3x² - 14x + 14; f'(c) = 3c² - 14c + 14 = 0; c = (7 ± √49 - 42)/3 = (7 ± √7)/3

c₁ = (7 - √7)/3 = 1.45 or c₂ = (7 + √7)/3 = 3.22