Q. 5
Verify Rolle’s theorem for each of the following functions:
Answer :
Condition (1):
Since, f(x)=x2-4x+3 is a polynomial and we know every polynomial function is continuous for all xϵR.
⇒ f(x)=x2-4x+3 is continuous on [1,3].
Condition (2):
Here, f’(x)=2x-4 which exist in [1,3].
So, f(x)=x2-4x+3 is differentiable on (1,3).
Condition (3):
Here, f(1)=(1)2-4(1)+3=0
And f(3)= (3)2-4(3)+3=0
i.e. f(1)=f(3)
Conditions of Rolle’s theorem are satisfied.
Hence, there exist at least one cϵ(1,3) such that f’(c)=0
i.e. 2c-4=0
i.e. c=2
Value of c=2 ϵ(1,3)
Thus, Rolle’s theorem is satisfied.
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