Verify Rolle’s theorem for each of the following functions:

Question

Philoid · Accepted Answer

Condition (1):

Since, f(x)=x²-x-12 is a polynomial and we know every polynomial function is continuous for all xϵR.

⇒ f(x)= x²-x-12 is continuous on [-3,4].

Condition (2):

Here, f’(x)=2x-1 which exist in [-3,4].

So, f(x)= x²-x-12 is differentiable on (-3,4).

Condition (3):

Here, f(-3)=(-3)²-3-12=0

And f(4)=4²-4-12=0

i.e. f(-3)=f(4)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one cϵ(-3,4) such that f’(c)=0

i.e. 2c-1=0

i.e.

Value of

Thus, Rolle’s theorem is satisfied.