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  "body": "\n## Statement\n\nThe [[exponential function|0RMVS7RXZX]] is strictly convex on $\\mathbb{R}$.\nThat is, $e^{t x + (1 - t) y} \u003c t e^x + (1 - t) e^y$ for all real $x \\ne y$ and $t \\in (0, 1)$.\n\n## Remarks\n\nThe second derivative of $e^x$ is [[$e^x$ itself|0EX9GFJ0B3]], which is positive. So [[the second derivative test for convexity|0Z41C17TGP]] applies, in its strict form.\nThe tangent line at $0$ gives the inequality [[$1 + x \\le e^x$|012C7GFF20]].\nMathlib states convexity as `convexOn_exp`.\n\n## Explanations\n\n### L1 human\n\nThe graph of $e^x$ bends upward everywhere.\nEvery chord of the graph lies above the graph.\n",
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