Section 4.3 Maxima and Minima
References.
- Calculus, Early Transcendentals by Stewart, Section 4.1.
Questions of optimization are one of the two most important applications of calculus that we will see this semester. (The other is finding a function from information about its rate of change, coming up in Section 4.10 and Chapter 5) For example, choosing the shape of a product that minimizes its weight or cost, or that maximizes strength for a given weight, or finding a route that minimizes travel time, or the price at maximizes profit.
We have already seen one key idea intuitively and graphically: the graph of function tends to have a low point or a high point where the derivative is zero, and not where the derivative is non-zero. We now investigate carefully questions like:
- When does a function have an overall minimum or maximum value?
- How can we find these extreme values of a function, and the arguments of the function that give them?
Try to draw a graph where the minimum value occurs at a point \(P(a,f(a))\text{,}\) where it is not true that \(f'(a)=0\text{.}\)
Checkpoint 4.3.2.
Try to draw a graph where \(f'(a)=0\) for some \(a\text{,}\) but the value \(f(a)\) is not a minimum or maximum, even compared to nearby points on the graph.
Definition 4.3.3. Global Extrema.
A function has an absolute maximum, or global maximum, at \(c\) if \(f(c) \geq f(x)\) for all \(x\) in its domain \(D\text{.}\) The value \(f(c)\) is the maximum value of \(f\) on domain \(D\text{.}\)
An absolute (global) minimum and minimum value are defined similarly. Collectively, global maxima and minima are global extrema, and the values of \(f\) there are extreme values.
Definition 4.3.4. Local Extrema.
A function has a local maximum, or relative maximum, at \(c\) if \(f(c) \geq f(x)\) for \(x\) near \(c\text{.}\) That is, on some open interval \((a,b)\) containing \(c\text{,}\) \(f(x)\) is never greater than \(f(c)\text{.}\)
The value of the function at a local maximum is a local maximum value.
Local (relative) minima, extrema and such are defined similarly.
Checkpoint 4.3.5.
Find the local [relative] and global [absolute] minima and maxima of the function \(f(x)=\cos x\text{,}\) and the points at which they occur.
Checkpoint 4.3.6.
Try to find the locations of the global extrema and corresponding extreme values for \(f(x)=x^2\text{.}\)
Checkpoint 4.3.7.
Do the same for \(g(x)=x^3\text{.}\)
Checkpoint 4.3.8.
Find the global extrema of \(f(x)=x^2\) with domain \(D=[-1,2]\text{.}\)
Find all local extrema for this function.
Checkpoint 4.3.9.
- Using a graph, try to find the local extrema for \(f(x)=3x^4-16x^3+18x^2\) on domain \(D=[-1,4]\text{.}\)
- Then find the global extrema of this function.
- Is the derivative of \(f\) zero at every local extremum?
- Where are the local extrema with \(f'(x) \neq 0\text{?}\)
Theorem 4.3.10. The Extreme Value Theorem.
If function \(f\) is continuous on a closed interval \([a,b]\text{,}\) then it attains a global maximum value \(f(c)\) and a global minimum value \(f(d)\) at some numbers \(c\) and \(d\) in this interval.
Note that either extreme value can possibly occur at more than one number, so that \(c\) and \(d\) are not always unique.
Theorem 4.3.11. Fermat's Theorem.
If function \(f\) has a local extremum on an open interval at number \(c\) and if \(f'(c)\) exists, then \(f'(c)=0\text{.}\)
- \(f'(c)\) might not exist at a local extremum.
- This refers to open intervals, so excluding endpoints: endpoints are also always candidate locations for extrema.
Checkpoint 4.3.12.
Use \(f(x)=x^3\) to show that having \(f'(c)=0\) does not always give a local extremum at \(x=c\text{.}\)
Checkpoint 4.3.13.
Use \(f(x)=|x|\) to show that not all local extrema on open intervals occur at points where \(f'(c)=0\text{.}\)
Checkpoint 4.3.14.
Use \(f(x)=x\) on domain \(D=[0,1]\) to show another way that local extrema can occur.
The above three exercises show that \(f'(x)=0\) is only part of the puzzle!
Definition 4.3.15. Critical Points, Values, and Numbers.
A critical number of a function \(f\) is a value \(c\) in its domain such that either \(f'(c)=0\) or \(f'(c)\) does not exist.
A critical value is the value \(f(c)\) of a function at a critical number \(c\text{.}\)
A critical point is a point \(P(c,f(c))\) on the graph of function \(f\) for \(c\) a critical number.
Loosely, a critical point is a point on the graph where the curve does not either rise or fall as it passes through the point, where rise or fall of the curve is indicated by a rising or falling tangent line.
Checkpoint 4.3.16.
Find the critical points of \(f(x)=x^{3/5}(x-4)\text{.}\)
Fermat's Theorem can now be rephrased this way:
Theorem 4.3.17. (Where the extrema are, and are not).
Local extrema can occur only at critical points and end points.
Thus, if \(f\) has a non-zero derivative at a point that is not an end point, that point does not give a local extremum.
Note: the “open interval” part of the original statement of Fermat's Theorem meant that it did not say anything about end points. Typically a function has only a finite number of critical points (and of end points), so once these are found, working out which of them give global minima or maxima is just a matter of computing and comparing the values at those points. This is a lot better than having to check at the infinite number of points in the domain of \(f\text{!}\)
Checking if a point is a local minimum or maximum sometimes requires a few more ideas, coming in the next few sections.
The Closed Interval Method for Finding Global Extrema.
The results above can be turned into a procedure for finding and classifying the extrema of a continuous function \(f\) on a closed bounded interval \([a,b]\text{.}\)
- Compute the derivative of \(f\text{,}\) and find where it is zero or does not exist, plus the end points.
- Find the value of \(f\) at each of these points.
- Compare these values: the largest and smallest are the global maximum and minimum of \(f\text{.}\)
Exercises Exercises
Study Calculus Volume 1, Section 4.3 2 ; in particular the Problem Solving Strategy, all Examples and Checkpoints, and a few Exercises from each of the ranges 91–98, 100–103, 104–107, 108–117, 118–128 and 129–134. (Some suggested selections are Exercises 91, 93, 97, 101, 107, 109, 119 and 129.)
openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minimaopenstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima
