Let f ( x ) = ln ( x 8 ) in the interval [1,6]. Determine the value(s) of 'c' that satisfy the Mean Value Theorem for this function in the given interval.

Question

Let  in the interval [1,6]. Determine the value(s) of 'c' that satisfy the Mean Value Theorem for this function in the given interval.

Answer 1

The MVT applies to any function continuous on a closed interval and differentiable on the open interval, both of which apply to the given function on [1 , 6].

The MVT guarantees at least one value c, with a < c < b , such that f^'(c) = [f(b) - f(a)] / [ b - a ]. In other words, at least one x-value in the interval where the instantaneous rate of change equals the average rate of change over the entire interval. Because f(x) is always increasing on this particular interval, there will only be 1 c value.

Btw, let's use a logarithm property to simplify f(x): f(x) = ln(x⁸) = 8lnx

f(6) = 8ln6 ; f(1) = 0 ; avg r o c = (8ln6) / 5

f^'(x) = 8 / x = (8ln6) / 5 ; x = 5 / ln6 so c = 5/ln6 .