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

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Calculus 2 / BC

Chapter 6

Applications of Integration

Section 5

Average Value of a Function

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

09:04

Use the diagram to show th…

02:32

$\begin{array}{l}{\text { …

03:08

a) if f and g are concave …

01:16

Prove that if $f$ and $g$ …

0:00

(a) If $f$ and $g$ are con…

02:33

Give an explanation for yo…

01:53

Let $f$ and $g$ represent …

01:08

If $f(x)$ is concave up, t…

02:40

01:06

Let $f$ and 8 represent di…

Show that the curve $y=f(x…

01:10

If $a>0, b>0,$ show …

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

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