Math Insight

Minimization and maximization refresher

The fundamental idea which makes calculus useful in understanding problems of maximizing and minimizing things is that at a peak of the graph of a function, or at the bottom of a trough, the tangent is horizontal. That is, the derivative $f'(x_o)$ is $0$ at points $x_o$ at which $f(x_o)$ is a maximum or a minimum.

Well, a little sharpening of this is necessary: sometimes for either natural or artificial reasons the variable $x$ is restricted to some interval $[a,b]$. In that case, we can say that the maximum and minimum values of $f$ on the interval $[a,b]$ occur among the list of critical points and endpoints of the interval.

And, if there are points where $f$ is not differentiable, or is discontinuous, then these have to be added in, too. But let's stick with the basic idea, and just ignore some of these complications.

Let's describe a systematic procedure to find the minimum and maximum values of a function $f$ on an interval $[a,b]$.

• Solve $f'(x)=0$ to find the list of critical points of $f$.
• Exclude any critical points not inside the interval $[a,b]$.
• Add to the list the endpoints $a,b$ of the interval (and any points of discontinuity or non-differentiability!)
• At each point on the list, evaluate the function $f$: the biggest number that occurs is the maximum, and the littlest number that occurs is the minimum.

Example 1

Find the minima and maxima of the function $f(x)=x^4-8x^2+5$ on the interval $[-1,3]$. First, take the derivative and set it equal to zero to solve for critical points: this is $$4x^3-16x=0$$ or, more simply, dividing by $4$, it is $x^3-4x=0$. Luckily, we can see how to factor this: it is $$x(x-2)(x+2)$$ So the critical points are $-2,0,+2$. Since the interval does not include $-2$, we drop it from our list. And we add to the list the endpoints $-1,3$. So the list of numbers to consider as potential spots for minima and maxima are $-1,0,2,3$. Plugging these numbers into the function, we get (in that order) $-2, 5, -11, 14$. Therefore, the maximum is $14$, which occurs at $x=3$, and the minimum is $-11$, which occurs at $x=2$.

Notice that in the previous example the maximum did not occur at a critical point, but by coincidence did occur at an endpoint.

Example 2

You have $200$ feet of fencing with which you wish to enclose the largest possible rectangular garden. What is the largest garden you can have?

Let $x$ be the length of the garden, and $y$ the width. Then the area is simply $xy$. Since the perimeter is $200$, we know that $2x+2y=200$, which we can solve to express $y$ as a function of $x$: we find that $y=100-x$. Now we can rewrite the area as a function of $x$ alone, which sets us up to execute our procedure: $$area = xy=x(100-x)$$ The derivative of this function with respect to $x$ is $100-2x$. Setting this equal to $0$ gives the equation $$100-2x=0$$ to solve for critical points: we find just one, namely $x=50$.

Now what about endpoints? What is the interval? In this example we must look at ‘physical’ considerations to figure out what interval $x$ is restricted to. Certainly a width must be a positive number, so $x>0$ and $y>0$. Since $y=100-x$, the inequality on $y$ gives another inequality on $x$, namely that $x <100$. So $x$ is in $[0,100]$.

When we plug the values $0,50,100$ into the function $x(100-x)$, we get $0,2500,0$, in that order. Thus, the corresponding value of $y$ is $100-50=50$, and the maximal possible area is $50\cdot 50=2500$.

Exercises

1. Olivia has $200$ feet of fencing with which she wishes to enclose the largest possible rectangular garden. What is the largest garden she can have?
2. Find the minima and maxima of the function $f(x)=3x^4-4x^3+5$ on the interval $[-2,3]$.
3. The cost per hour of fuel to run a locomotive is $v^2/25$ dollars, where $v$ is speed, and other costs are $100 per hour regardless of speed. What is the speed that minimizes cost per mile?
4. The product of two numbers $x,y$ is 16. We know $x\geq 1$ and $y\geq 1$. What is the greatest possible sum of the two numbers?
5. Find both the minimum and the maximum of the function $f(x)=x^3+3x+1$ on the interval $[-2,2]$.