Extrema of Functions¶
We have seen how to differentiate functions from several different families, now we can use this knowledge to help us understand the maxima and minima of functions.
Extrema of Functions¶
If we say \(x = a\) is an extremum of a function \(f\), this means that \(f\) has a maximum or minimum at \(x = a\). (The plural of "extremum" is "extrema", of course.)
Definition¶
The basic idea of a maximum is that it occurs when the function value is highest:
Let's say that a
global maximum(or absolute maximum) occurs at \(x = a\) if \(f(a)\) is the highest value of \(f\) on the entire domain of \(f\);local maximum(or relative maximum) occurs at \(x = a\) if \(f(a)\) is the highest value of \(f\) on some small interval containing a.
Every global maximum is also a local maximum.
The Extreme Value Theorem¶
Let's say that \(x = c\) is a critical point for the function \(f\) is either \(f'(c) = 0\) or \(f'(c) = DNE\). Then we have the Extreme Value Theorem:
Suppose that \(f\) is defined on \((a,b)\) and \(c\) is in \((a, b)\). If \(c\) is a local maximum or minimum of \(f\), then \(c\) must be a critical point for \(f\). That is, either \(f'(c) = 0\) or \(f'(c) = DNE\).
How to Find Global Maxima and Mimima¶
In glory detail, here's how to find the global maximum and minimum of the function \(f\) with domain \([a, b]\):
- Find \(f'(x)\). Make a list of all the points in \((a, b)\) where \(f'(x) = 0\) or \(f'(x) = DNE\). That is, make a list of all the critical points in the interval \((a, b)\).
- Add the endpoints \(x = a\) and \(x = b\) to the list.
- For each of the points in the list, find the y-coordinates by substituting into the equation \(y = f(x)\).
- Pick the highest y-coordinate and note all the values of x from the list coresponding to that y-coordinate. These are the global maxima.
- Do the same for the lowest y-coordinate to find the global minima.
Rolle's Theorem¶
Suppose that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). If \(f(a) = f(b)\), then there must be at least one number \(c\) in \((a, b)\) such that \(f'(c) = 0\).
The Mean Value Theorem¶
Suppose that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). Then there's at least one number \(c\) in \((a, b)\) such that:
The Mean Value Theorem looks a lot like Rolle's Theorem. In fact, the conditions for applying the two theorems are almost the same. In both cases, your function \(f\) has to be continuous on a closed interval \([a, b]\) and differentiable on \((a, b)\). Rolle's Theorem also requires that \(f(a) = f(b)\), but Mean Value Theorem doesn't require that. In fact, if you apply the Mean Value Theorem to a function \(f\) statisfying \(f(a) = f(b)\), you'll see that \(f(b) - f(a) = 0\), so you get a number \(c\) in \((a, b)\) statisfying \(f'(c) = 0\). So Mean Value Theorem reduces to Rolle's Theorem.