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.) c = Based off of this information, what conclusions can be made about the Mean Value Theorem? This contradicts the Mean Value Theorem since f satisfies the hypotheses on the given interval but there does not exist any c on (2, 5) such that f '(c) = f(5) − f(2) 5 − 2 . This does not contradict the Mean Value Theorem since f is not continuous at x = 3. This does not contradict the Mean Value Theorem since f is continuous on (2, 5), and there exists a c on (2, 5) such that f '(c) = f(5) − f(2) 5 − 2 . This contradicts the Mean Value Theorem since there exists a c on (2, 5) such that f '(c) = f(5) − f(2) 5 − 2 , but f is not continuous at x = 3. Nothing can be concluded.

          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.)
c = 
Based off of this information, what conclusions can be made about the Mean Value Theorem?
This contradicts the Mean Value Theorem since f satisfies the hypotheses on the given interval but there does not exist any c on (2, 5) such that f '(c) = 
f(5) − f(2)
5 − 2
.
This does not contradict the Mean Value Theorem since f is not continuous at x = 3.    
This does not contradict the Mean Value Theorem since f is continuous on (2, 5), and there exists a c on (2, 5) such that f '(c) = 
f(5) − f(2)
5 − 2
.
This contradicts the Mean Value Theorem since there exists a c on (2, 5) such that f '(c) = 
f(5) − f(2)
5 − 2
, but f is not continuous at x = 3.
Nothing can be concluded.

Added by Jasmin T.

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