Finding Intervals of Monotonicity and Local Extrema using First Derivative Test

Chapter 4.3: Finding intervals of monotonicity using first derivative of a function, First
Derivative Test for Local Extrema
Based on technical lemma; if f’(x0) will be positive then the relation of the derivative for a point
x that will be nearby though smaller than x0 will imply that the value of f(x) to be less than f(x0).
A different approach will be applied when the x point is nearby but larger than the x0, which will
mean that the value of f(x) will be larger than f(x0).
Proof: the technical lemma proof which involves defining the derivative and definition of the
limit.
Suppose the derivative f’(x0) = p, this will mean that
lim
x→ x 0
f ( x ) −f (x 0)
x−x 0 = p
By taking x close to x0 the value of f ( x ) −f (x 0)
x −x 0 will be approximately close to p
Therefore, x > x0 if and only if f(x) > f(x0).. This will imply that if x is near x0 but greater than x0,
then f(x) will be greater than f(x0) and vice versa is true.
It will follow that if f will take an extreme value at point x0 in an interval that is open, then the f
derivative can neither be negative or positive and hence f will have a derivative of zero. The
conclusion is only applicable at the interior of an interval.
Rolle’s Theorem: A function will have a maximum and minimum value and the derivative will
be zero if the extrema occurs at the interior point. But if the extrema will occur at the end point
then the function will be a constant.