Final answer:
To find the numbers b such that the average value of f(x)= 2+10x-9x^2 on the interval [0,b] is equal to 3, we need to solve an integral and set it equal to 3. The integral can be solved using the given equation and the value(s) of b can be found by solving a cubic equation.
Step-by-step explanation:
To find the numbers b such that the average value of f(x)= 2+10x-9x^2 on the interval [0,b] is equal to 3, we need to find the average value of the function on that interval and set it equal to 3. The average value of the function on an interval is given by the formula: Average value = (1/(b-0)) * ∫[0,b] (2+10x-9x^2) dx. Setting this equal to 3, we can solve the integral and find the value of b.
Using the given equation, the integral becomes: Average value = (1/b) * [2x + 5x^2 - 3x^3] evaluated from 0 to b. Setting this equal to 3 and solving for b, we get b^3 - 5b^2 + 6b - 6 = 0. This is a cubic equation that can be solved to find the value(s) of b.
Once we have the value(s) of b, we can substitute it back into the original equation to check if the average value is indeed equal to 3.