Consider the function f(x)=x^3-3x^2+2x-6 on the interval [0,2].

Question

Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval;

f(x) is continuous on [0,2]

f(x) is differentiable on (0,2)

and f(0)=f(2)= -6

Then by Rolle's theorem, there exists a c such that f'(c)=0

Find the values c that satisfy the conclusion of Rolle's theorem.

c1= ____and c2=____ with c1<c2

Answer 1

Differentiation of the given function with respect to x yields

f'(x) = d/dx (x³ - 3x² + 2x - 6) = 3x² - 6x + 2

Setting f'(c) = 0, we have

f'(c) = 3c² - 6c + 2 = 0

3(c² - 2c) + 2 = 0

3(c² - 2c + 1 - 1) + 2 = 0

3(c² - 2c + 1) - 3 + 2 = 0

3(c - 1)² - 1 = 0

(c - 1)² = 1/3

c = 1 ± √(1/3)

Since it is given that c₁ < c₂, we must have

c₁ = 1 - √(1/3) ≈ 0.423

c₂ = 1 + √(1/3) ≈ 1.577

So, 0 < c₁ < c₂ < 2.