MAGIX Music Maker Soundpool DVD Collection MEGA PACK 9 – 19

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MAGIX Music Maker Soundpool DVD Collection MEGA PACK 9 – 19 World. chaecedowi. 2018. 7. 18. 00:12 ëŒ“ê¸€ìˆ˜0 ê³µê°ìˆ˜0.
MAGIX Music Maker Soundpool DVD Collection MEGA PACK 9 – 19 Watch Online World.chaecedowi. 2019. 7. 18. 05:20 ëŒ“ê¸€ìˆ˜0 ê³µê°ìˆ˜0.Q:

Total differential of a function not zero.

Suppose $f$ is $2$ times differentiable on $[a,b]$ and $\int_a^b f(x) dx=0$. Can we conclude that $f(a)=f(b)=0$?
Intuitively it seems that we can because the sum of the two values over the entire domain is $0$.

I guess you mean that $f$ is twice differentiable. Let us assume $f$ continuously differentiable. Then, $f(a)=f(b)$ means that $f$ attains its extremum at $a$ and $b$ and the differential of $f$ must be zero at both those points. Moreover, we have $\displaystyle\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}=0$ and $\displaystyle\lim_{h \to 0} \frac{f(b-h)-f(b)}{h}=0$ which means that the derivative of $f$ is continuous at $a$ and $b$. From what you’ve written, we get $f(x)=0$ for all $x$, which contradicts the fact that the integral is zero.

The claim is false as made, i.e. if $f$ is $2$ times differentiable, then $f(a) = f(b)$ implies that $f

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