The MVT links the average rate of change and the instantaneous rate of change
There must be some point in the interval where the slope of the tangent line equals the slope of the secant line (that connects the endpoints)
Rolle’s Theorem is a special case of the MVT
It means that a continuous, differentiable curve has a horizontal tangent between any two points
Take the derivative of the function
Set it equal to zero to find your critical numbers
Plug in a number above the critical point and below the critical point to find the sign of f’(x)
To find the absolute (global) extrema you have to consider the endpoints and critical numbers
Make a table of these values
Then plug back into your ORIGINAL FUNCTION
The first derivative tells us if the function is increasing or decreasing
The second derivative tells us if this is happening at an increasing or decreasing rate
This is given to us by something called concavity
Take the second derivative of the function
Set it equal to zero to find your points of inflection
Plug in a number above the critical point and below the critical point to find the sign of f”(x)