Short and Sweet Calculus

4.2Extreme Values of Functions

• In many practical problems, we need to find the greatest (maximum) value or the least (minimum) value—there can be more than one of each—of a function.

• The maximum and minimum values of a function are collectively called the extreme values or extrema of the function.

singular form plural form
maximum maxima
minimum minima
extremum extrema
• Differentiation can help us locate the extreme values of a function.

• In calculus, there are two types of “maximum” and “minimum,” which are distinguished by the two prefixes: absolute (also called global) and relative (also called local). The adjective “local” refers to a comparison with nearby values, while “global” or “absolute” refers to a comparison with all values.

• The absolute maximum (or global maximum) of a function is the greatest value of the function.

• The absolute minimum (or global minimum) of a function is the least value of the function.

• A local maximum (or relative maximum) of a function is a value that is greater than all values immediately preceding and following.

• A local minimum (or relative minimum) of a function is a value that is less than all values immediately preceding and following.

• Geometrically speaking, local maxima and local minima are respectively the “peaks” and “valleys” of the curve. The global maximum is highest point and the absolute minimum is the lowest point of the curve.

Figure 4.3 shows the graph of function $$f$$ with absolute maximum at $$x=b$$ and absolute minimum at $$x=p$$. The point $$(c,f(c))$$ is higher than all nearby points on the curve, although it is not as high as $$(b,f(b))$$. That is, if we consider only values of $$x$$ sufficiently close to $$c$$, then $$f(c)$$ is the largest value of those values of $$f$$. In this case, we say $$f(c)$$ is a local maximum or $$f$$ has a local maximum at $$x=c$$.

Similarly, we say $$f$$ has a local (or relative) minimum at $$x=d$$ because $$f(x)\geq f(d)$$ for all $$x$$ very close to $$d$$. Other local maxima and minima are denoted on Figure 4.3.6

• At a local maximum, the graph of the function makes the transition from rising to falling. At the peak, if the tangent line exists, the tangent line must be horizontal; that is, the slope of the tangent line must be zero.

For example, in Figure 4.3, the tangent lines are horizontal at $$x=c$$ and $$x=p$$. At $$x=e$$, the graph has a corner, and thus the tangent line does not exist.

• Similarly, at a local minimum, the graph of the function makes the transition from falling to rising. At the valley, if the tangent line exists, the tangent line must be horizontal; that is, its slope must be zero.

For example, in Figure 4.3, the tangent lines are horizontal at $$x=d$$ and $$x=p$$. At $$x=s$$, the graph has a cusp, and the tangent line does not exist.

• From the above observations, we conclude that if $$f$$ has a local maximum or minimum at $$x=c$$, then $$f'(c)=0$$ (when the slope of the tangent line is zero) or $$f'(c)$$ does not exists (when there is no tangent line).

4.2. (Fermat’s Theorem): If $$f(c)$$ is a local maximum or minimum, then either $$f'(c)=0$$ or $$f'(c)$$ does not exist.

• A number in the domain of a function $$f$$ at which the derivative is zero or the derivative does not exist has a special name. It is called a critical number or a critical point. The value of $$f$$ at a critical number is called a critical value.

• Fermat’s theorem says that if $$f(c)$$ is a local maximum or a local minimum, then $$c$$ is a critical number of $$f$$.

• It is important to understand that every local maximum or minimum is a critical value, but a critical value is not necessarily a local maximum or a minimum. For example, in Figure 4.3, the tangent line is horizontal at $$x=r$$ or $$f'(r)=0$$, but $$f(r)$$ is not an extreme value of $$f$$. On the left of this point, the curve is falling (meaning the function is decreasing); at $$x=r$$, the curve momentarily flattens out, and on its right, it starts falling again. Therefore, we can every single local extreme value is a critical value, but not every critical value is necessarily a local extreme value.

Finding the Absolute Extrema of a Function

Obviously every absolute extreme value, with the exception of an absolute extreme value that occurs at an endpoint, is also a local extreme value. Hence:

• An absolute maximum or minimum of a function occurs either at a critical point or at an endpoint of its domain.

• This provides us a method to find the absolute maximum and the absolute minimum of a continous function on a finite closed interval $$[a,b]$$.

Strategy for finding absolute extrema of a continuous function $$f$$ on a finite closed interval $$[a,b]$$:

• Step 1: Find $$f'(x)$$

• Step 2: Find all critical values: Set $$f'(x)=0$$ and solve it for $$x$$. Also find every value of $$x$$ for which $$f'(x)$$ does not exist. Evaluate $$f$$ at each of these numbers that lie between $$a$$ and $$b$$.

• Step 3: Evaluate $$f(a)$$ and $$f(b)$$.

• Step 4: The largest value of $$f$$ from Steps 2 and 3 is the absolute maximum of $$f$$ and the least value of $$f$$ from these steps is the absolute minimum of $$f$$ on $$[a,b]$$.

Example 4.2. Find the absolute maximum and minimum value of the function $f(x)=\frac{1}{3}x^{3}-4x$ on the interval $$[-3,4]$$.

Solution

Step 1: Finding the derivative of $$f$$: $f(x)=\frac{1}{3}x^{3}-4x\Rightarrow f'(x)=x^{2}-4$ Step 2: Finding the critical values of $$f$$. The function is differentiable everywhere, so all the critical numbers are obtained by setting $$f'(x)=0$$: $f'(x)=x^{2}-4=0\Longleftrightarrow x=\pm2.$ Because both $$x=2$$ and $$x=-2$$ lie between $$-3$$ and $$4$$, we evaluate $$f$$ at both of these numbers: $f(2)=\frac{1}{3}(2^{3})-4(2)=-\frac{16}{3}\approx-5.333,$ $f(-2)=\frac{1}{3}(-2)^{3}-4(-2)=\frac{16}{3}\approx5.333.$ Step 3: Evaluating $$f$$ at the endpoints $f(-3)=\frac{1}{3}(-3)^{3}-4(-3)=3,\qquad f(4)=\frac{1}{3}(4^{3})-4(4)=\frac{16}{3}\approx5.33.$ Step 4: Comparing the critical values and the endpoint values.

$$x$$ $$-3$$ $$-2$$ $$2$$ $$4$$
$$f(x)$$ $$3$$ $$\frac{16}{3}$$ $$-\frac{16}{3}$$ $$\frac{16}{3}$$
max min max

The absolute maximum of $$f$$ on $$[-3,4]$$ is $$16/3$$, which occurs at $$x=-2$$ and $$x=4$$, and its absolute minimum on this interval is $$-16/3$$, which occurs at $$x=-2$$. The graph of $$f$$ is shown in Figure 4.4.

1. We do not say that $$f$$ has a local maximum at $$x=b$$ or a local minimum at $$x=a$$ because there is no point immediately following $$x=b$$ and no point immediately preceding $$x=a$$. However, some textbooks compare the height at these two points with the existing immediately near points.↩︎