Definition
Exponential Tangent Inequality
For every
, the natural exponential function lies above its tangent line at : Equality holds if and only if
. Equivalently, replacing by gives
Proof
Proof
Let
. Since
is strictly convex. A differentiable convex function lies above each of its tangent lines: Taking
yields Strictly convexity makes equality possible only at the tangency point
. Substituting for yields .