Final answer:
To find the value of c in terms of a that satisfies the Mean Value Theorem, we use the formula for average rate of change and set up an equation. By simplifying and solving for c, we get c = b - 2a.
Step-by-step explanation:
To find the value of c in terms of a that satisfies the Mean Value Theorem for the function f(x) = ax/(a-x), we need to use the formula for the average rate of change of a function.
According to the Mean Value Theorem, there exists a value c in the interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a).
We can start by finding the derivative of f(x):
f'(x) = [a(a-x) - ax]/(a-x)^2
= -a^2/(a-x)^2
Now we can set up the equation for the average rate of change:
-a^2/(a-c)^2 = (f(b) - f(a))/(b - a)
Substituting f(a) = (a*a)/(a-a)
= a, f(b)
= (b*a)/(b-a)
= -a*b/(b-a),
we get:
-a^2/(a-c)^2 = (-a*b/(b-a) - a)/(b - a)
Cross multiplying and simplifying, we find:
a*(b-a-c) = -a*b
Canceling out the a terms, we have:
b - a - c = -b
Solving for c, we get:
c = b - 2a