 Minimax linear approximation of a square-root endpoint profile

On $[0,1]$, the chord of $\sqrt{1-x}$ is $1-x$. Its nonnegative gap has maximum $1/4$ at $x=3/4$, as differentiation shows. Shifting the chord upward by $1/8$ gives error $-1/8,1/8,-1/8$ at $0,3/4,1$. The Chebyshev alternation theorem therefore proves that $9/8-x$ is the unique best linear uniform approximation, with error $1/8$. Endpoint differentiability is unnecessary.