The Mean Value Theorem and the Comparison Theorem imply that
A) for and .
B) for and .
C) for and .
D) for and .
E) None of the above.

Question

The Mean Value Theorem and the Comparison Theorem imply that
A) for and . 
B) for and . 
C) for and . 
D) for and . 
E) None of the above.

Quizplus · Accepted Answer

The Mean Value Theorem states that for a function f continuous on [a, b] and differentiable on (a, b), there exists a point c in (a, b) such that f'(c) = (f(b) - f(a))/(b-a). The Comparison Theorem states that if g(x) ≤ f(x) for all x in [a, b], then g'(x) ≤ f'(x) for all x in (a, b).

Using these theorems, we can see that for choice B, the function with the largest derivative of the three is 11eaa655_3b2c_bd57_a696_8f5da3734037_TB5596_11, since the other two have values that are less than or equal to it. Therefore, by the Comparison Theorem, it has the largest derivative.

The Mean Value Theorem and the Comparison Theorem Imply That