Counterexamples of second mean value theorem of integral Recently I am having trouble understanding Bonnet's mean value theorem. I looked back and forth on the proof, and find that the condition of $\phi$ being decreasing is essential. However, I am unable to find an example to illustrate how the theorem fails when $\phi$ is NOT decreasing. 

Can anyone give an example?
DO NOT LEAVE a comment before even reading the question. The question is asking for an example where the theorem FAILS. It is NOT asking for an example where the theorem applies properly. It is irresponsible to vote or comment before even understanding the question properly.

Let $\phi(x)=x$, $G(x)=1$, and $[a,b]=[0,1]$. Then
$$
\int_0^1 G(x)\phi(x)dx=0.5
$$

But for any $c\in[0,1]$, 
$$
\phi(0) \int_0^c G(x)dx=0.
$$
So the theorem fails.
DO NOT LEAVE a comment before even reading the question. The question is asking for an example where the theorem FAILS. It is NOT asking for an example where the theorem applies properly. It is irresponsible to vote or comment before even understanding the question properly.

Let $\phi(x)=x$, $G(x)=1$, and $[a,b]=[0,1]$. Then
$$
\int_0^1 G(x)\phi(x)dx=0.5
$$

But for any $c\in[0,1]$, 
$$
\phi(0) \int_0^c G(x)dx=0.
$$
So the theorem fails.