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#2: Post edited
- >Since $F(x)$ is continuous in the closed interval [a,a+h] and differentiable in the open interval (a,a+h). Also f(a) = f(b) so by Rolle's theorem we get
- $$F'(a+\theta h)=0,a<a+\theta h<a+h$$
As you seen in the quoted description (I took it from my book) $f(a)=f(b)$ But, I saw in the same book in my language they wrote that $F(a)=F(a+h)$ instead of $f(a)=f(b)$. I think it's a mistake. But, I can't understand which one is correct. Even, I can't see any derivation of the equation $F'(a+\theta h)=0,a<a+\theta h<a+h$
- >Since $F(x)$ is continuous in the closed interval [a,a+h] and differentiable in the open interval (a,a+h). Also f(a) = f(b) so by Rolle's theorem we get
- $$F'(a+\theta h)=0,a<a+\theta h<a+h$$
- As you seen in the quoted description (I took it from my book) $f(a)=f(b)$ But, I saw in the same book in my language they wrote that $F(a)=F(a+h)$ instead of $f(a)=f(b)$. I think it's a mistake. But, I can't understand which one is correct. Even, I can't see any derivation of the equation $F'(a+\theta h)=0,a<a+\theta h<a+h$
- <hr/>
- <details>
- <summary>book</summary>
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- </details>
- <hr/>
- Let $$F(x)=f(x)+(a+h-x)f'(x)+A(a+h-x)^2$$-----1
- $$=>F(a)=f(a)+(h)f'(a)+A(h)^2$$--------2
- Differentiate (1) respect to $x$ :
- $$F'(x)=f'(x)-f'(x)+(a+h-x)f''(x)-2A(a+h-x)$$
- $$=>F'(x)=(a+h-x)f''(x)-2A(a+h-x)$$
- $$=>F'(a+\theta h)=(h-\theta h)f''(h+\theta h)-2A(h-\theta h)$$
- As we seen, from that book $F'(a+\theta h)=0$.
- So,
- $$=>2A(h-\theta h)=(h-\theta h)f''(h+\theta h)$$
- $$=>A=\frac{f''(h+\theta h)}{2}$$
- Put the value in (1). Then, you will get second mean value theorem. Sorry! I can't edit it now for shortness of time.
#1: Initial revision
Second mean value theorem proof (differentiation)
>Since $F(x)$ is continuous in the closed interval [a,a+h] and differentiable in the open interval (a,a+h). Also f(a) = f(b) so by Rolle's theorem we get $$F'(a+\theta h)=0,a<a+\theta h<a+h$$ As you seen in the quoted description (I took it from my book) $f(a)=f(b)$ But, I saw in the same book in my language they wrote that $F(a)=F(a+h)$ instead of $f(a)=f(b)$. I think it's a mistake. But, I can't understand which one is correct. Even, I can't see any derivation of the equation $F'(a+\theta h)=0,a<a+\theta h<a+h$