Suppose we have a convex function $f(x)$ and a point $A=(x_1,y_1)$, where $y_1 \leq f(x)$. How to find a tangent line to $f(x)$ from point $A$?
(I'm not sure whether we can get a closed-form expression.)
2026-04-12 05:06:54.1775970414
How to find a tangent line to a convex function from a point.
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The tangent line at $(a,b=f(a))$ is $y-b=f'(a)(x-a)$ assuming that $f$ is differentiable. This has to pass through $A$ so $y_1-f(a)=f'(a) (x_1-a)$. We can only define $a$ implicitly by this equation. We cannot get a closed form for $(a,b)$.