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- #1

\[

\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.