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) = x3 − x2 − 6x + 1, [0, 3]

Answer 1

1. f is continuous

2. f is differentiable

3. f(0)= 1
f(3)= 1
three conditions are satisfied.


$\frac{dy}{dx} = 3 {x}^{2} - 2x - 6$
now
$\frac{dy}{dx} = 0$
$x = \frac{2 ( + )( - ) \sqrt{4 + 72} }{6}$

$= 2( + )( - ) 6\sqrt{2} \div 6$
$= \frac{1 + 3 \sqrt{2} }{3} .. \frac{1 - 3 \sqrt{2} }{3}$
The required value of c is
$\frac{1 + 3 \sqrt{2} }{3}$