Thread: Mean Value Theorem

Let f:R--->R twice differentiable on R. Suppose that f(a)=f(b)=f(c) for real numbers a<b<c. show that there exists cE(a,c) such that

do I apply Mean Value Theorem to the interval (a,c)? why do i need the function to be twice differentiable to answer the question?

thanks

Originally Posted by charikaar

Let f:R--->R twice differentiable on R. Suppose that f(a)=f(b)=f(c) for real numbers a<b<c. show that there exists cE(a,c) such that

do I apply Mean Value Theorem to the interval (a,c)? why do i need the function to be twice differentiable to answer the question?

thanks

since is differentiable, and , then by the MVT, there exists an such that ... same argument for an such that

since f is twice differentiable, then the MVT also applies to the function , and since , then there exists a value such that

Originally Posted by charikaar

Let f:R--->R twice differentiable on R. Suppose that f(a)=f(b)=f(c) for real numbers a<b<c. show that there exists cE(a,c) such that

do I apply Mean Value Theorem to the interval (a,c)? why do i need the function to be twice differentiable to answer the question?

Apply Rolle's theoremto show that the first derivative f' has a zero in each of the intervals (a,b) and (b,c). Then apply Rolle's theorem again, this time to the function f', to show that f" has a zero in the interval (a,c).