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If you can differentiate the function, then you can tell that the graph is concave down if the second derivative is negative over the range examined. As an example: for f(x) = -x2, f'(x) = -2x and f"(x) = -2 < 0, so the function will be everywhere concave down.

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Q: How do you determine if the graph of a function is concave down without looking at the graph?
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