I already did this and the answer was 185 b Findcsuch thatfa

(I already did this and the answer was 18/5)

b) Findcsuch thatfave=f(c). (Enter solutions from smallest to largest. If there are any unused answer boxes, enter NONE in the last boxes. Round the answers to three decimal places.)

c1=

c2=

Solution

f_ave=Integration ( f(x)dx) / Integration (dx)

=Integration (18x/(1+x²)²)dx /2

Let 1+x²=z

So, 2xdx=dz

or, 18xdx=9dz

So,

f_ave=integration ( 9dz/z²)/2

=(-9/z) /2

=(-9/(1+x²))/2

Putting limits x=0 to 2

=(-9/5 + 9/1)/2

=(36/5)/2

=18/5

For f_ave=f(c),

f(c)=18/5

or, 18c/(1+c²)²=18/5

or, 5c = (1+c²)²

or, c⁴+2c²-5c+1=0

Solving, you get

c=0.22,1.21