Question

Find the numbers b such that the average value of f(x) = 4 + 10x − 6x2 on the interval [0, b] is equal to 5.

Answer 1

Answer:

  b = (5±√17)/4 ≈ {0.2192, 2.2808}

Step-by-step explanation:

The average on the interval is found from the integral ...

  $\frac{1}{b}\int\limits^b_0 {(4+10x-6x^2)} \, dx =\frac{1}{b}(4b+5b^2-2b^3)\\\\=-2b^2+5b+4$

We want the value of this integral to be 5, so we want to find b such that ...

  5 = -2b^2 +5b +4

  2b^2 -5b +1 = 0

  b^2 -5/2b = -1/2

  b^2 -5/2b +25/16 = -1/2 +25/16

  (b -5/4)^2 = 17/16

  b = (5±√17)/4

The average value of f(x) is 5 on the intervals ...

  [0, (5-√17)/4] and [0, (5+√17)/4]