Can any twice-differentiable function satisfy these conditions?

[Dustinsfl]

I want to show for all twice differentialable functions there are no functions that take a min or max with the following conditions \(y(-1) = -1\), \(y(1) = 1\), and
\[
\int_{-1}^1x^2y^{'2}dx.
\]

From the E-L eq, we have \(2x^2y' = c\).
So
\[
y(x) = -\frac{c}{6} + d
\]
Using the conditions, we get
\[
\begin{pmatrix}
\frac{1}{6} & 1 & -1\\
-\frac{1}{6} & 1 & 1
\end{pmatrix}\Rightarrow
\begin{pmatrix}
\frac{1}{6} & 1 & -1\\
0 & 2 & 0
\end{pmatrix}
\]
Therefore, we have an inconsistent system since no two tuples satisfy the bottom equation.
Thus, no twice differentiable function with the given conditions takes on a minimum or maximum.

 

[Ackbach]

I think from $2x^{2}y'=c$, the next step would be
$$y'= \frac{c}{2x^{2}}=(c/2)x^{-2} \quad \implies \quad y=- \frac{c}{2x}+d.$$

 

[Dustinsfl]

I think from $2x^{2}y'=c$, the next step would be
$$y'= \frac{c}{2x^{2}}=(c/2)x^{-2} \quad \implies \quad y=- \frac{c}{2x}+d.$$

Not everyone can integrate but after that error, the result analysis would be the same correct?

 

[Ackbach]

If you just apply the BC's, you might have
\begin{align*}
1&=-c+d\\
-1&=c+d.
\end{align*}
From here, you could say that $d=0$ and $c=-1$, and hence $y=1/x$. The problem is that this function is not in the collection of functions in which you were interested - at least not on the interval $[-1,1]$. That is, $1/x$ is not twice-differentiable on $[-1,1]$.

 

[CellCoree]

I would like to point out that the statement "for all twice differentiable functions there are no functions that take a min or max with the following conditions" is not entirely accurate. While it is true that the specific function described in the content does not satisfy the given conditions, it does not necessarily mean that there are no other twice differentiable functions that could satisfy those conditions. It is possible that there are other functions that could satisfy the given conditions, but they would have to be different from the one described in the content. Therefore, it would be more accurate to say that the function described in the content does not satisfy the given conditions, rather than making a general statement about all twice differentiable functions. Additionally, it is important to consider the domain and range of the function in question, as it could also affect the ability of the function to satisfy the given conditions.