Find all numbers c that satisfy the conclusion of the Mean Value Theorum6th root of x **or (x) ^1/6**, [0,1]?

Question

please show steps
Mean Value theorum states
f(b) - f(a)= f'(c)(b-a)

Answer 1

in this case:
a = 0
b= 1

if (f(x) = x^(1/6)
f(a) = 0
f(b) = 1

so you want to find all numbers c such that:

(1 - 0) = f'(c)(1 - 0) or
f'(c) = 1

The derivative of x^n is nx^(n-1)

The above applies even when n is not an integer. In this case, n = 1/6, so

f'(c) = (1/6)x^(-5/6) = (1/6)/(x^(5/6))

So the equation you need to solve is:

x^(5/6) = 1/6

If you need an approximation to a numerical solution, take the log of both sides to get a linear equation, solve the linear equation for ln(x), and exponentiate to get an approximation to x.

If you need more accuracy, use the above as a starting value for an iteration.

I think you would also find it helpful to plot f(x) and f'(x), just to get a feel for how they behave, and how many values of c you need to compute.