Find c in the given interval

Question

Consider the following given function and given interval.

g(x) = (x + 2)³, [0, 2]

Find c in the given interval such that g_ave = g(c).

Answer 1

The average value of the function across a range of x is equal to the integral normalized by the length of the range. In math that means:

g_ave = ∫_xi^xfg(x) dx / (x_f - x_i)

First let's take the integral of the provided function:

∫g(x) = 1/4 (x + 2)⁴ (1)

Applying the above formula for average value using the bounds of [0, 2]:

g_ave = [(1/4 (2 + 2)⁴) - (1/4 (0 + 2)⁴)] / (2 - 0) = [64 - 4] / 2 = 30

Now we want to find the value of c such that g(c) = g_ave

30 = (c + 2)³

3√30 = c + 2

c = 3√30 - 2 or ~1.1072

Answer 2

g'(c) = g(2) - g(0) /(2-0), Mean Value Theorem.

Find g'(x) and solve for x.