Suppose $f: \mathbb R \longrightarrow \mathbb R$ is strictly convex and twice differentiable. Fix $a \in [0,1]$ and let $$g(x) = (a-x)f''(x).$$ Can we say anything about monotonicity properties of $g$ over the unit interval? In particular, is $g$ monotonically decreasing over $[0,1]$?
2026-03-26 11:00:10.1774522810
Is $(a-x)f''(x)$ decreasing for $f$ convex?
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