#### To determine

**To show:**

By using the diagram, if f is a concave upward on a,b then

fave>fa+b2

#### Answer

If f is a concave upward on a,b then

fave>fa+b2

#### Explanation

**1) Concept:**

Use the mean value theorem for integrals.

**2) Mean Value Theorem for Integrals**

If f is continuous on a,b then there exists a number c in a,b such that

fc=fave=1b-a∫abfx dx

that is,

∫abfx dx=fcb-a

**3) Calculation:**

f is concave upward on a,b

The area under the curve f on a,b is

∫abfx dx.

From the graph, the area under the curve f on a,b contains a trapezoid ABDF.

Therefore,

∫abfx dx>area of trapezoid ABDF

By using the mean value theorem of the integral,

f is continuous on a,b

fave=1b-a∫abfx dx

>1b-aarea of trapezoid ABDF

From the graph,

a+b2 is midpoint of a and b.

Therefore, AG=GF, CH=HE

We know ∠CHB=∠EHD and ∠HCB=∠HED.

Therefore, ∆CHB≅∆EHD

So, area of trapezoid ABDF=area of rectangle ACFE

So,

fave=1b-a∫abfx dx

>1b-aarea of trapezoid ABDF

=1b-aarea of rectangle ACFE

From the graph, the width of rectangle is (b-a) and height of the rectangle is

fa+b2

So, the

area of rectangle ACFE=width*lenght=fa+b2*b-a

=1b-afa+b2*b-a

=fa+b2

Therefore,

fave>fa+b2

**Conclusion:**

If f is concave upward on a,b, then

fave>fa+b2.