Definition 4.2.1.
Broadly, a relative maximum is a point \(x = a\) such that \(f(a) \geq f(x)\) for all \(x\) near \(a\text{.}\) Similarly, a relative minimum is a point \(x = a\) such that \(f(a) \leq f(x)\) for all \(x\) near \(a\text{.}\)
Section 4.2 Local Extrema of Functions
A large application of calculus is for finding the maximums and minimums of functions.
Subsection 4.2.1 Local/Relative Extrema
The singular of extrema is extremum (the word originates from Latin). An extremum refers to either a maximum or minimum.
Note that there some math textbooks have the convention that allows endpoints of the domain of a function to be possible local extrema. Whereas, other textbooks do not, requiring that a function be defined on both sides of its local extrema.
Subsection 4.2.2 Critical Points
Let \(f\) be a smooth function. Fermat recognized that maxima and minima corresponded with horizontal tangents, where the slope, or derivative, is equal to 0. Intuitively, a function \(f\) can only have a local maximum if it is a point \(x = a\) where the derivative exists and is equal to 0. More generally, the function can also have a sharp corner or cusp, where the derivative is not defined. Then, we call points like this critical points.
Definition 4.2.2.
Let \(f\) be defined at \(x = a\text{.}\) Then, \(a\) is a critical point of \(f\) if either \(f'(a) = 0\) or \(f'(a)\) does not exist (\(f\) is not differentiable at \(x = a\)).
This definition requires that \(f\) be defined at its critical numbers, so that critical points correspond to points on the graph of \(f\text{.}\) For example, \(f(x) = \frac{1}{x}\) does not have a critical number at \(x = 0\text{.}\)
Subsection 4.2.3 Fermat's Theorem (Interior Extremum Theorem)
Theorem 4.2.3. Fermat's theorem.
Let \(f\) be differentiable at \(x = a\text{.}\) Then, if \(f(a)\) is a local extremum, then \(f'(a) = 0\text{.}\) Equivalently, if \(f'(a) \neq 0\text{,}\) then \(f(a)\) is not a local extremum of \(f\text{.}\)
Fermat's theorem provides a necessary condition for determining local extrema. In other words, if we determine all of the points where \(f'(a) = 0\text{,}\) then this provides a list of candidate points for extrema to occur. Points \(x\) where \(f'(x) = 0\text{,}\) or \(f'(x)\) doesn't exist but \(f\) does, are candidates for local extrema
In this way, determining possible extrema involves solving the equation \(f'(x) = 0\text{.}\) Note that the converse of Fermat's theorem is false, in that it is possible for \(f'(a) = 0\text{,}\) but \(f\) does not have a local extremum at \(x = a\text{.}\)
Example 4.2.4.
Subsection 4.2.4 First Derivative Test
Consider a differentiable function \(y = f(x)\text{.}\) From the graph of such functions, we can observe that a function cannot change from increasing to decreasing, without first going over a “hill” where the derivative is zero. Similarly, the graph of a function cannot change from decreasing to increasing without going through a “valley”. Together, we conclude that if the slope is never zero, then the graph can't change directions.
Theorem 4.2.5.
Let \(f\) be continuous. If \(f\) has no critical points on an interval \(I\text{,}\) then \(f\) is either always increasing on \(I\) or always decreasing on \(I\text{.}\)
More precisely, if the derivative \(f'\) is continuous, then by the intermediate value theorem, if it is positive at one point and negative at a different point, then it must be 0 somewhere between those two points. Then, if it is never 0, then it must be either always positive or always negative.
In this way, the critical points, and the points where the function is not defined, provide markings for intervals where the function can change from increasing to decreasing, or vise versa.
Theorem 4.2.6. First derivative test.
Let \(f\) be a continuous function, \(a\) be a critical point of \(f\text{.}\) Then,
- If \(f'\) changes from positive to negative at \(a\text{,}\) then \(f\) has a local maximum at \(a\text{.}\)
- If \(f'\) changes from negative to positive at \(a\text{,}\) then \(f\) has a local minimum at \(a\text{.}\)
- If \(f'\) is positive both to the left and right at \(a\text{,}\) or is negative both to the left and right at \(a\text{,}\) then \(f\) has neither a local minimum nor a local maximum at \(a\text{.}\)
Subsection 4.2.5 Endpoints as Local Extrema (Advanced)
If we use the convention that endpoints can be local extrema, then we can also use insight from the derivative to determine if a function has a local max or local min at the endpoints of an interval.
Example 4.2.7.
For the function \(f(x) = \sqrt{x}\text{,}\) it has a local minima at \(x = 0\text{.}\) However, the derivative \(f'(x) = \frac{1}{2\sqrt{x}}\text{.}\) Notice that as \(x \to 0+\text{,}\) \(f'(x) \to \infty\text{.}\)
Let \(f\) be a function, differentiable on \([a,b]\text{.}\) Then,
- \(x = a\) is a local maximum (minimum) if and only if,\begin{equation*} \lim_{x \to a+} f'(x) \lt 0 \qquad \brac{\text{respectively, } \lim_{x \to a+} f'(x) > 0 \end{equation*}
- \(x = b\) is a local maximum (minimum) if and only if,\begin{equation*} \lim_{x \to b-} f'(x) > 0 \qquad \brac{\text{respectively, } \lim_{x \to b-} f'(x) \lt 0 \end{equation*}
Subsection 4.2.6 Remarks
Note that there is a difference between a maximum value and the point at which it is achieved. The maximum is the output of the function which is largest; the location of the maximum is the input of the function at which the maximum occurs. Sometimes, maximum point refers to the coordinate pair \((x,y)\) of the maximum. The terminology is somewhat mixed.
Technically, it is not true in general that functions which are differentiable have continuous derivatives. However, functions which fail to have continuous derivatives are quite pathological, and not relevant in any applications.
Subsection 4.2.7 Advanced Examples
Example 4.2.8.
Prove that \(\tan{x} > x\) for \(x \in (0,\pi/2)\text{,}\) by showing that \(f(x) = \tan{x} - x\) is increasing on \((0,\pi/2)\text{.}\)
In general, if \(f(x) = g(x) - h(x)\) satisfies \(f(a) \geq 0\text{,}\) and \(f'(x) > 0\) on the interval \((a,b)\text{,}\) then \(g(x) > h(x)\) on \((a,b)\text{.}\)
Subsection 4.2.8 Fermat's Method of Adequality (Historical Note)
In 1629, Fermat developed a technique of finding extremum of functions, later published in his treatise Methodus ad disquirendam maximam et minimam (Method of Finding Maxima and Minima). This was before the development of the derivative.
- Let \(f(x) = f(x + h)\text{.}\)
- Simplify the equation, and cancel like terms.
- The remaining terms will contain a common factor of \(h\text{.}\) Cancel out this common factor.
- Discard terms which involve \(h\text{,}\) i.e. set \(h = 0\text{.}\)
- Solve for \(x\)
Implicitly, this divided by \(h\text{,}\) then sets \(h\text{,}\) and has the similar paradox of the derivative. It is equivalent to the derivative method.
