# Function's value at midpoint of interval and concavity

\frac{f(a)+f(b)}{2}\ge f\left(\frac{a+b}{2}\right)

I do not have the exact question with me right now but it had 4 options

I drew such a diagram for concave up and down.

I do not know where to place (f(a) + f(b))/2 in comparison to the f(mid point of an and b).

I am sorry for the multiple images. There was an error in the latex code. [Edit: Now fixed.]

Actually, itâ€™s great being able to see your work since we can tell some of what youâ€™re already thinking.

Building off of what you have, hereâ€™s one way to proceed: Draw the straight line through the points \big(a, f(a)\big) and \big(b, f(b)\big), and see first if that helps you draw any conclusions. And then if you need another hint, tap to unhide this:

Notice that the the midpoint of that line segment lies at \big(\frac{a + b}{2}, \frac{f(a) + f(b)}{2}\big).

I imagine that if you compare whatâ€™s going on at the midpoint of that line to what happens with the different concavities, youâ€™ll be able to draw the conclusion youâ€™re after.

Hope that helps, and please let us know if that generates further questions for you.

Lovely hint. You are a wonderful teacher. Just drawing that line helped me come to a conclusion.
I am assuming that drawing a line helps us analyse whether the curve goes above or below the line.

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Yes, thatâ€™s exactly how the line helps here. In fact, Calculus ends up using lines a lot, including developing approximations (â€ślinear approximationâ€ť = replacing a short segment of a functionâ€™s curve with a line instead), something that often isnâ€™t emphasized nearly enough in beginning courses.

Glad the hint(s) helped, and thank you very much for the compliment. Much appreciated!!

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