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) = 4x2 − 8x + 3, [−1, 3] c =

Answer 1

Answer with explanation:

Rolle's Theorem states that:

If f is a continuous function in [a,b] and is differentiable in (a,b)

such that f(a)=f(b)

Then there exist a constant c in between a and b i.e. c∈[a,b]

such that:  f'(c)=0

Here we have the function f(x) as:

$f(x)=4x^2-8x+3$ where x∈[-1,3]

• Since the function f(x) is a polynomial function hence it is continuous as well as differentiable over the interval [-1,3].

Also,

f(-1)=15

(Since,

$f(-1)=4\times (-1)^2-8\times (-1)+3\\\\f(-1)=4+8+3\\\\f(-1)=15$   )

and f(3)=15

( Since,

$f(3)=4\times 3^2-8\times 3+3\\\\f(3)=36-24+3\\\\i.e.\\\\f(3)=15$ )

• i.e. f(-1)=f(3)

Hence, there will exist a c∈[-1,3] such that f'(c)=0

$f'(x)=8x-8\\\\i.e.\\\\f'(x)=0\\\\imply\\\\8x-8=0\\\\i.e.\\\\x-1=0\\\\i.e.\\\\x=1$

Hence, the c that satisfy the conclusion is: c=1