MHB Can any twice-differentiable function satisfy these conditions?

  • Thread starter Thread starter Dustinsfl
  • Start date Start date
AI Thread Summary
No twice-differentiable function can satisfy the conditions \(y(-1) = -1\), \(y(1) = 1\), and the integral condition involving \(y'\). The analysis using the Euler-Lagrange equation leads to an inconsistent system, indicating that no such function can achieve a minimum or maximum under these constraints. Attempts to derive a function result in \(y = -\frac{c}{2x} + d\), but this form does not yield a valid twice-differentiable function on the interval \([-1, 1]\). Specifically, the function \(y = 1/x\) fails to meet the twice-differentiability requirement within the specified interval. Consequently, the conclusion is that no twice-differentiable function exists that meets the given criteria.
Dustinsfl
Messages
2,217
Reaction score
5
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.
 
Mathematics news on Phys.org
I think from $2x^{2}y'=c$, the next step would be
$$y'= \frac{c}{2x^{2}}=(c/2)x^{-2} \quad \implies \quad y=- \frac{c}{2x}+d.$$
 
Ackbach said:
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?
 
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 $[-1,1]$.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top