Problem 24 Hard Difficulty

Use the diagram to show that if $ f $ is concave upward on $ [a, b] $, then
$$ f_{ave} > f \left(\frac{a + b}{2}\right) $$

Answer

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Video Transcript

we know the definition of average value one over B minus eight times the integral from A to B F of x d. X. Therefore, what we know is that the integral represents the area under the blue curve. Therefore, be minus a times off of a plus two b over to Is this some of the rectangles that are black? We know the area under the blue curve is greater than the area off the triangles. Put together some of the area of the rectangles put together. Therefore, we know that we can write the average so a V E means average is greater than you want. Us eight times be mine, say times off of a post be divided by two. Therefore, we know that substituting in the integral we end up with simply the average is greater than half of a post. Be divided by two. Just remember to substitute the intro back in