Consider a differentiable function $u : [ 0,1 ] \rightarrow \mathbb { R }$. Assume the function $u$ satisfies $$u ( a ) = \frac { 1 } { 2 r } \int _ { a - r } ^ { a + r } u ( x ) d x , \quad \text{for all } a \in ( 0,1 ) \text{ and all } r < \min ( a , 1 - a ).$$ Which of the following four statements must be true? (A) $u$ attains its maximum but not its minimum on the set $\{ 0,1 \}$. (B) $u$ attains its minimum but not maximum on the set $\{ 0,1 \}$. (C) If $u$ attains either its maximum or its minimum on the set $\{ 0,1 \}$, then it must be constant. (D) $u$ attains both its maximum and its minimum on the set $\{ 0,1 \}$.
Consider a differentiable function $u : [ 0,1 ] \rightarrow \mathbb { R }$. Assume the function $u$ satisfies
$$u ( a ) = \frac { 1 } { 2 r } \int _ { a - r } ^ { a + r } u ( x ) d x , \quad \text{for all } a \in ( 0,1 ) \text{ and all } r < \min ( a , 1 - a ).$$
Which of the following four statements must be true?\\
(A) $u$ attains its maximum but not its minimum on the set $\{ 0,1 \}$.\\
(B) $u$ attains its minimum but not maximum on the set $\{ 0,1 \}$.\\
(C) If $u$ attains either its maximum or its minimum on the set $\{ 0,1 \}$, then it must be constant.\\
(D) $u$ attains both its maximum and its minimum on the set $\{ 0,1 \}$.