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$f''(x) < 0$ and $f'(0) = 0$. Then, is the sum $f(0) + 2f(2) + 2f(4) + f(6)$ larger or smaller than $f$'s integral from $0$ to $6$?

Does the answer change when we have $f(x) > 0$ for $x \geq 0$?

Attempt

I thinks the answer should be "cannot be determined with the available info", even with the extra condition. Because the sum could be smaller than the lower Riemann on the partition with points $0,2,4,6$, or it could be less than the upper Riemann sum on the same partition.

2 Answers 2

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Hint: The sum is the result of integrating with the trapezoidal rule. The function is concave. Draw a picture to see where this leads you.

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    Ah yesss! That was the key insight. Can't believe I didn't see it. Thanks. +12012-11-01
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Since $f''(x) < 0$ and $f'(0) = 0$ it is not possible to have $f(x) > 0$ for all $x \geq 0$, but in any case the value of f(0) makes no difference to the required calculations to answer your general question.

A sketch would look like

enter image description here

and if you take the excesses and deficits resulting from your approximation in three pairs, it is easy to see that the concave curve means that each excess is smaller than the following deficit and so the approximation is smaller than the integral. Now you know the answer, you can prove it analytically.