Find all values of c

Question

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 1

f(5) = 1/4

f(2) = 1

the slope of the secant line is [ f(5) - f(2) ] / (5-2)

= (1/4 - 1) / 3

= (-3/4)/3

= -1/4

f'(x) = -2(x-3)^(-3)

MVT says there exists c in [2,5] such that

-2(c-3)^(-3) = -1/4

1/(c-3)^3 = 1/8

cross mutliplies:

8 = (c-3)^3

8 = (c-3)^3

2 = c-3

c = 5