Let f(x) = (x − 3)−2. Find all values of c in (2, 5) such that f(5) − f(2) = f '(c)(5 − 2). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Answer:Step-by-step explanation:This is the Mean Value Theorem for Calculus.  The formula for that, at least the one I use for it, resembles yours, but I don't simplify it down quite as far, as I find it easier to work with:[tex]f'(c)=\frac{f(b)-f(a)}{b-a}[/tex]That says that the derivative of the function at some value of c within a < c < bis equal to the slope of the function on the interval.We need f'(c), f(b) and f(a) to solve this.  First the derivative.  I have to say that I am assuming the function is[tex]f(x)=(x-3)^{-2}[/tex]If so, the derivaitve of this function is[tex]f'(x)=\frac{-2}{x-3}[/tex]Next, f(b).  We plug in a 5 for x in the function to get that f(b) = .25Next, f(a).  We plug in a 2 for x in the function to get that f(a) = 1Filling in the MVT:[tex]\frac{-2}{x-3}=\frac{.25-1}{5-2}[/tex]Cross multiply to get-.75(x - 3) = -6 and-.75x + 2.25 = -6 so-.75x = -8.25 andx = 11 (that is the same as c = 11).  Now we can find the y value that corresponds to that x value by subbing 11 into x in the function:[tex]f(11)=(11-3)^{-2}[/tex][tex]f(11)=\frac{1}{64}[/tex]Therefore, the value of c where the slope of the function is the same as the slope of the derivative is at (11, 1/64)