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Thread: Mean Value Theorem

  1. #1
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    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 $\displaystyle f''(d)=0$

    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
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    Quote Originally Posted by charikaar View Post
    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 $\displaystyle f''(d)=0$

    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 $\displaystyle f$ is differentiable, and $\displaystyle f(a) = f(b)$ , then by the MVT, there exists an $\displaystyle x_1 \in (a,b)$ such that $\displaystyle f'(x_1) = 0$ ... same argument for an $\displaystyle x_2 \in (b,c)$ such that $\displaystyle f'(x_2) = 0$

    since f is twice differentiable, then the MVT also applies to the function $\displaystyle f'(x)$ , and since $\displaystyle f'(x_1) = f'(x_2)$ , then there exists a value $\displaystyle d \in (x_1,x_2)$ such that $\displaystyle f''(d) = 0$
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    Quote Originally Posted by charikaar View Post
    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 $\displaystyle f''(d)=0$

    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).
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