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  "body": "\n## Proof\n\nWe show that $f'$ is increasing and then use the derivative test for convexity.\n\n1. Since $f$ is twice differentiable on $I$, the function $f' \\colon I \\to \\R$ is differentiable on $I$. Its derivative is the [[second derivative|0Y9D2FJRQ4]] $f''$.\n2. A differentiable function is [[continuous|07ZMX4ANJS]], so $f'$ is continuous on $I$.\n3. Every point of the open interval $I$ is an interior point of $I$, and $(f')'(x) = f''(x) \\ge 0$ there.\n4. By the [[nonnegative derivative test|026BJBQNGJ]] applied to $f'$, the function $f'$ is increasing on $I$.\n5. The function $f$ is differentiable on $I$ and $f'$ is increasing. By the [[derivative test for convexity|005JT76HCK]], $f$ is [[convex|0VV0HSW1ZS]] on $I$.\n\n## Idea\n\nA nonnegative second derivative means the first derivative never decreases. So the slope of $f$ only grows from left to right, and a function whose slope grows is convex.\n",
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