Find the average value Brooklyn Math GQS

Question

Find the average value of the function f(x)=3x²-6x on the interval [1,3],

and find value of c in this interval such that f(c) equals the average value.

Answer 1

The average value of an integrable function ƒ on the interval [a, b] is

1/(b-a)∫f(x)dx where integral is from a to b

in your case calculate:

1/(3-1)∫3x^2-6xdx where integral is from 1 to 3

For the second part you just solve f(c)= average value you found above

Answer 2

fav = 1/(3 - 1)∫1³(3x² - 6x)dx = 1/2(x³ - 3x²)₁³ = 1/2((27 - 27) - (1 - 3)) =1;

1 = 3c² - 6c; 3c² - 6c - 1 = 0; c = (3 ± 2√3)/3 = 1 ± 2√3/3.

Only c = 1 + 2√3/3 from interval [1, 3]