Determine if Rolle’s Theorem can be applied. If Rolle’s Theorem can be applied, find all values of c in the interval such that

Question

Please show as much work as possible. Thank you!

Determine if Rolle’s Theorem can be applied. If Rolle’s Theorem can be applied, find all values of c in the interval such that f'(c) = 0, f (x) = (x + 3)(x + 2) ² closed interval [-3, 2] 

Answer 1

For Rolle's Theorem to apply, f(a) must equal f(b) on the interval [a, b] and the function must be differentiable and continuous on (a, b).

For [-3, 2], does f(-3) = f(2)? If not, Rolle's Theorem is not applicable,

Answer 2

f(x) = (x+3)(x+2)^2

f(-3) =(-3+3)(-3+2)^2 = 0(1) = 0

f(2) = (2+3)(2+2)^2 = 5(16) = 80

f'(x) = (x+3)(2) + (x+2) = 2x+6+x+2 = 3x+8

f'(c) = 3c+ 8 = (80-0)/(2+3) = 16

3c+8 =16

c= (16-8)/3

c = 8/3

Rolle's Theorem doesn't apply, but the Mean Value Theorem does apply giving

c=8/3

Rolles' Theorem is just the Mean Value Theorem with one little added restriction