ACUERDO DEL CONSEJO GENERAL DEL INSTITUTO ELECTORAL DEL ESTADO DE QUERÉTARO, POR EL QUE SE ORDENA AL DIRECTOR GENERAL A REALIZAR Y CONTINUAR CON LAS MODIFICACIONES

4.4. Curve sketching

Activity 4.6 Sketch the curve y = f (x) where f (x) = x²e^x and find all of its points of inflection.

4.4.3 Asymptotes and cusps

The method above for sketching y = f (x) assumes, as we generally have throughout this chapter, that the function, f (x), and its derivatives are well-defined for all x∈ R.

But, more generally, there may be points at which the function or some of its

derivatives are not defined. When this happens we start to encounter asymptotes and cusps. We will not dwell on this a great deal here, but we can use the following examples to see how this may affect our sketches.

Example 4.10 Sketch the curve y = (x− 1)⁻¹.

Here we have y = f (x) where the function, f (x), is given by f (x) = 1

x− 1, as long as x6= 1. In particular, this means that we have

f⁰(x) = − 1

(x− 1)² and f⁰⁰(x) = 2 (x− 1)³,

and so these derivatives aren’t defined at x = 1 either.² Using these, we can see that when

x < 1 we have f (x) < 0, f⁰(x) < 0 and f⁰⁰(x) < 0, meaning that for these values of x the function is negative, decreasing and concave; whereas when

x > 1 we have f (x) > 0, f⁰(x) < 0 and f⁰⁰(x) > 0, meaning that for these values of x the function is positive, decreasing and convex.

We can also see that the y-intercept of this curve occurs when y =−1 and that f (x)→ 0 as x → ±∞ which means that this function has a horizontal asymptote given by y = 0. However, the main feature that concerns us here is the vertical asymptote at x = 1 which comes about because

xlim→1⁻f (x) =−∞ and lim

x→1⁺f (x) =∞,

as we should expect to see from our discussion of hyperbolae in Section 2.2.4. The sketch of this curve is illustrated in Figure 4.12(a).

In particular, observe that in Example 4.10, we have a case like the one mentioned at the end of Section 4.3.3. That is, the function changes from being concave to convex at a point, but there is no point of inflection. This happens because the second derivative of this function does not exist at the point.

2That is, the function and its derivatives are undefined when x = 1 as that would require us to ‘divide by zero’ and that is never allowed.

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Example 4.11 Sketch the curve y = (x− 1)⁻².

Here we have y = f (x) where the function, f (x), is given by f (x) = 1

(x− 1)², as long as x6= 1. In particular, this means that we have

f⁰(x) = − 2

(x− 1)³ and f⁰⁰(x) = 6 (x− 1)⁴,

and so these derivatives aren’t defined at x = 1 either.³ Using these, we can see that when

x < 1 we have f (x) > 0, f⁰(x) > 0 and f⁰⁰(x) > 0, meaning that for these values of x the function is positive, increasing and convex; whereas when

x > 1 we have f (x) > 0, f⁰(x) < 0 and f⁰⁰(x) > 0, meaning that for these values of x the function is positive, decreasing and convex.

We can also see that the y-intercept of this curve occurs when y = 1 and that f (x)→ 0 as x → ±∞ which means that this function has a horizontal asymptote given by y = 0. However, the main feature that concerns us here is the vertical asymptote at x = 1 which comes about because

x→1lim⁻f (x) =∞ and lim

x→1⁺f (x) =∞,

as now, f (x) is always positive. The sketch of this curve is illustrated in Figure 4.12(b).

Example 4.12 Sketch the curve y = (x− 1)^2/3.

Here we have y = f (x) where the function, f (x), is given by f (x) = (x− 1)^2/3,

which is defined for all x∈ R. However, this means that we have f⁰(x) = 2

3(x− 1)^1/3 and f⁰⁰(x) = − 2 9(x− 1)^4/3,

and so these derivatives aren’t defined at x = 1.⁴ Using these, we can see that when x < 1 we have f (x) > 0, f⁰(x) < 0 and f⁰⁰(x) < 0, meaning that for these values of x the function is positive, decreasing and concave; whereas when

3Again, the function and its derivatives are undefined when x = 1 as that would require us to ‘divide by zero’ and that is never allowed.

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4.5. Optimisation

x > 1 we have f (x) > 0, f⁰(x) > 0 and f⁰⁰(x) < 0, meaning that for these values of x the function is positive, increasing and concave.

We can also see that the y-intercept of this curve occurs when y = 1. The sketch of this curve is illustrated in Figure 4.12(c) and we say that this curve has a cusp at x = 1.

−1 y

x

x = 1

y = 1 x− 1 O

1 O x y

x = 1

y = 1

(x− 1)²

1

1 y

O x

y = (x− 1)^2/3

(a) (b) (c)

Figure 4.12: Sketches of the curves in (a) Example 4.10, (b) Example 4.11 and (c) Example 4.12. Observe the behaviour of all three of these curves at x = 1: in (a) and (b) we have a vertical asymptote at x = 1 and in (c) we have a cusp at x = 1.

4.5 Optimisation

We have seen how to use derivatives to find and classify the stationary points of a function and we have seen that a local maximum (or local minimum) is a point where the function is larger (or smaller) than it is at other nearby points. However, we now want to find the points, called a global maximum (or global minimum), where the function is larger (or smaller) than it is at all other points. In such cases, we often say that we are looking for the points where the function is optimised. We will see that some functions do not have a global maximum (or a global minimum) even though they may have a local maximum (or a local minimum).

In order to determine whether a function, f (x), has a global maximum or a global minimum, it is always useful to ask the following questions.

Which local maximum gives the largest value of f (x) and which local minimum gives the smallest value of f (x)?

What is the behaviour of f (x) as x→ ∞ and as x → −∞?

Then, having answered these questions one should be in a position to identify the global maximum with the largest value of f and the global minimum with the smallest value of f assuming, of course, that these exist. Indeed, one way of making sense of these questions and their answers is to sketch the relevant features of the curve y = f (x) and then, using this sketch, one can then easily identify any global maximum or global minimum that the function may have.

4We can see that these derivatives are undefined when x = 1 as that would require us to ‘divide by zero’ and that is never allowed. Moreover, observe that this function does not have a vertical tangent line at x = 1 because to the left of x = 1 the gradient is tending to−∞ and to the right of x = 1 the gradient is tending to∞.

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For instance, consider the function whose graph is sketched in Figure 4.13(a) which has two local maxima and two local minima. If we ask our questions about this function, we see that:

Comparing the relevant values, we see that the largest local maximum occurs when x = a and the smallest local minimum occurs when x = b.

The function tends to zero as x→ ±∞.

So, in this case, it should be clear that the global maximum occurs when x = a and the global minimum occurs when x = b as illustrated in Figure 4.13(b). However, if we have

y

O x

local max

local min

local min local max

b

a

y

O x

global max

local max

global min b

a

local min

(a) The sketch (b) The identification

Figure 4.13: (a) A sketch of a function with two local maxima and two local minima which tends to zero as x → ±∞. (b) This function has a global maximum and a global minimum as indicated.

the function sketched in Figure 4.14(a) and ask our questions about that we see that:

Comparing the relevant values, we see that the largest local maximum occurs when x = a and the smallest local minimum occurs when x = b.

The function tends to zero as x→ −∞ but tends to −∞ as x → ∞.

In this case, as illustrated in Figure 4.14(b), it should be clear that the global maximum still occurs when x = a but now there is no global minimum since we can get far smaller values of the function as x→ ∞ than we do from the smallest local minimum.

Activity 4.7 Use the sketches in Figures 4.9(b), 4.10(b) and 4.11(b) to determine whether the functions in Examples 4.7, 4.8 and 4.9 have any global maxima or global minima.

So, in general, we can see that if f : R→ R is a function that is differentiable for all x∈ R, then

its global maximum (or global minimum) can exist if the function is suitably well-behaved as x → ∞ and x → −∞; and

if they exist, its global maximum (or global minimum) must occur at a local maximum (or a local minimum).

But, having said this, a sketch is still the easiest way to see what is happening. We now turn to some cases of optimisation where things work slightly differently.