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Some people might use the second derivative as the definition of a concave function. But I think the intuitive definition is that graph is above the line joining two points of the graph for arbitrary two points.

Although, By pictorially, it seems that two definitions are equivalent. BUT how to prove that two are?

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1 Answer

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let $x_1 \leq x_2$ and $\alpha \in [0,1]$. Since we know that $f''(x_1)\leq 0$ if and only if $\frac{f(x_2) - f(x_1)}{x_2 - x_1}\leq \frac{f(y) - f(x_1)}{y - x_1}$ where $y = (1-\alpha)x_1 + \alpha x_2$.

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