F(x)=x 3−9xf, left parenthesis, x, right parenthesis, equals, x, cubed, minus, 9, x What is the average rate of change of f over the interval [2,6] open bracket, 12, comma, 6, close bracket?

Step-by-step explanation:

Given

Required

Determine the average rate of change

Average rate is calculated as thus;

In this case; b = 12 and a = 6

Calculating f(12)

Calculating f(6)

Substitute 12 for b and 6 for a in

Substitute values for f(6) and f(12)

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4