How many values of c is the derivative of the function equal to its average rate of change?

[onesun0000]

my first guess to the question was 4 because there would be 4 tangent lines parallel to the secant connecting the two end points. Then my second guess is none. No value of c can make derivative of the function equal to its average rate of change because no tangent line will be parallel to the secant and besides there is a sharp part of the graph where the function is not differentiable. I am not sure what the correct answer is.



A. Zero
B. Two
C. Three
D. Four

 

[Dr.Peterson]

Can you explain why you say, "no tangent line will be parallel to the secant", having said that you see 4 such places? Are you thinking of a theorem that contradicts your eyes, or what?

 

[onesun0000]

Can you explain why you say, "no tangent line will be parallel to the secant", having said that you see 4 such places? Are you thinking of a theorem that contradicts your eyes, or what?

Well, I thought of the Mean Value Theorem. The given function is continuous over the closed interval [a,b] but is not differentiable on the open interval (a,b) because of the sharp part of the graph. That's why I thought there would be no such value of c because the functions fails to satisfy the Mean Value Theorem.

 

[Dr.Peterson]

That's what I wondered.

The MVT doesn't say that there is such a tangent line ONLY when the function is differentiable; it says that when the function is differentiable, THEN there is such a tangent line. It can still exist under other conditions.

You are reading the MVT backward. It's like if you knew that if it rains, the grass will be wet, and concluded that since it didn't rain, the grass can't be wet. (Maybe I turned on the sprinkler.)

 

[onesun0000]

That's what I wondered.

The MVT doesn't say that there is such a tangent line ONLY when the function is differentiable; it says that when the function is differentiable, THEN there is such a tangent line. It can still exist under other conditions.

You are reading the MVT backward. It's like if you knew that if it rains, the grass will be wet, and concluded that since it didn't rain, the grass can't be wet. (Maybe I turned on the sprinkler.)

Oh yeah. MVT only says that a value c at which the derivative of a function is equal to the ARC if the function is differentiable on the open interval, as well as continuous on the closed interval, but never says it won't exist at all. Now it makes me more confused as to how many values can c be on the question.

 

[Dr.Peterson]

Oh yeah. MVT only says that a value c at which the derivative of a function is equal to the ARC if the function is differentiable on the open interval, as well as continuous on the closed interval, but never says it won't exist at all. Now it makes me more confused as to how many values can c be on the question.

Your initial thought was right. Just draw the four parallel tangent lines on the graph and you see they exist.

The theorem only says there is at least one such point under certain conditions; it doesn't deny the possibility of more than one, or under different conditions. That is probably the point of the exercise: to exorcise wrong impressions about the theorem!