Theorems

Question 1

Rolle’s Theorem

Answer 1

If f(x) is continuous on [A,B] and is differentiable on (a,b), and f(a) = f(b),
Then there exists a point c where the derivative f’(c) is equal to zero.

Question 2

Mean Value Theorem

Answer 2

If f(x) is continuous on [A,B]
and f(x) is differentiable on (a,b)

Then there exists a c such that the slope of the tangent line at c is equal to the secant of the interval

Question 3

First Derivative Theorems

Answer 3

If f’(x) is positive on an interval, then the function f(x) is increasing on that interval, by MVT. Same for negatives.