Consider the function f(x)=x^3-3x^2+2x-18 on the interval [0,2].

Question

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

f(x) is ____ on [0,2];

f(x) is ____ on (0,2);

and f(0) =f(2)= ______

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

Continuous on [0,2], differentiable on (0,2) and f(0)=f(2)=-18

At x=1+sqrt(3)/3, f(x)=0

Answer 2

f(x) = x³ - 3x² + 2x - 18

As an algebraic function the polynomial, f(x), is continuous on x = [0,2].

f(0) = 0 - 0 + 0 - 18 = -18

f(2) = 2³ -3(2²) + 2(2) - 18 = 8 - 12 + 4 - 18 = -18

Because f(0) = f(2) = -18, according to Rolle's theorem, there exists at least one value x=c

in the interval [0,2] such that f'(c) = 0.

To find c, we should find the derivative of f(x) and set it to zero.

f '(x) = 3x² - 6x + 2 = 0

Solve the quadratic equation to obtain

x = (1/6) [6 ± √(36 - 24) ]

= (6 ± √12)/6

= 1.5774, or 0.4226

Both values of x belong to (0, 2).

Answer:

c₁ = 0.4226 and c₂= 1.5774