Q. 3

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

Answer :

Condition (1):

Since, f(x)=x^{2}-5x+6 is a polynomial and we know every polynomial function is continuous for all xϵR.

⇒ f(x)= x^{2}-5x+6 is continuous on [2,3].

Condition (2):

Here, f’(x)=2x-5 which exist in [2,3].

So, f(x)= x^{2}-5x+6 is differentiable on (2,3).

Condition (3):

Here, f(2)=2^{2}-5×2+6=0

And f(3)= 3^{2}-5×3+6=0

i.e. f(2)=f(3)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one cϵ(2,3) such that f’(c)=0

i.e. 2c-5=0

i.e.

Value of

Thus, Rolle’s theorem is satisfied.

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