**MAXIMA AND MINIMA,** in mathematics. By the *maximum*
or *minimum* value of an expression or quantity is meant primarily
the “greatest” or “least” value that it can receive. In general,
however, there are points at which its value ceases to increase and
begins to decrease; its value at such a point is called a maximum.
So there are points at which its value ceases to decrease and
begins to increase; such a value is called a minimum. There
may be several maxima or minima, and a minimum is not
necessarily less than a maximum. For instance, the expression
(*x*^{2} + *x* + 2)/(*x* − 1) can take all values from −∞ to −1 and
from +7 to +∞, but has, so long as *x* is real, no value between
-1 and +7. Here −1 is a maximum value, and +7 is a
minimum value of the expression, though it can be made
greater or less than any assignable quantity.

The first general method of investigating maxima and minima
seems to have been published in A.D. 1629 by Pierre Fermat.
Particular cases had been discussed. Thus Euclid in book III.
of the *Elements* finds the greatest and least straight lines that can
be drawn from a point to the circumference of a circle, and in
book VI. (in a proposition generally omitted from editions of his
works) finds the parallelogram of greatest area with a given
perimeter. Apollonius investigated the greatest and least
distances of a point from the perimeter of a conic section, and
discovered them to be the normals, and that their feet were the
intersections of the conic with a rectangular hyperbola. Some
remarkable theorems on maximum areas are attributed to
Zenodorus, and preserved by Pappus and Theon of Alexandria.
The most noteworthy of them are the following:—

1. Of polygons of n sides with a given perimeter the regular polygon encloses the greatest area.

2. Of two regular polygons of the same perimeter, that with the greater number of sides encloses the greater area.

3. The circle encloses a greater area than any polygon of the same perimeter.

4. The sum of the areas of two isosceles triangles on given bases, the sum of whose perimeters is given, is greatest when the triangles are similar.

5. Of segments of a circle of given perimeter, the semicircle encloses the greatest area.

6. The sphere is the surface of given area which encloses the greatest volume.

Serenus of Antissa investigated the somewhat trifling problem of finding the triangle of greatest area whose sides are formed by the intersections with the base and curved surface of a right circular cone of a plane drawn through its vertex.

The next problem on maxima and minima of which there
appears to be any record occurs in a letter from Regiomontanus
to Roder (July 4, 1471), and is a particular numerical example
of the problem of finding the point on a given straight line at
which two given points subtend a maximum angle. N. Tartaglia
in his *General trattato de numeri et mesuri* (*c.* 1556) gives, without
proof, a rule for dividing a number into two parts such that
the continued product of the numbers and their difference is a
maximum.

Fermat investigated maxima and minima by means of the
principle that in the neighbourhood of a maximum or minimum
the differences of the values of a function are insensible, a method
virtually the same as that of the differential calculus, and of
great use in dealing with geometrical maxima and minima. His
method was developed by Huygens, Leibnitz, Newton and others,
and in particular by John Hudde, who investigated maxima and
minima of functions of more than one independent variable, and
made some attempt to discriminate between maxima and minima,
a question first definitely settled, so far as one variable is concerned,
by Colin Maclaurin in his *Treatise on Fluxions* (1742).
The method of the differential calculus was perfected by Euler
and Lagrange.

John Bernoulli’s famous problem of the “brachistochrone,” or curve of quickest descent from one point to another under the action of gravity, proposed in 1696, gave rise to a new kind of maximum and minimum problem in which we have to find a curve and not points on a given curve. From these problems arose the “Calculus of Variations.” (See Variations, Calculus of.)

The only general methods of attacking problems on maxima
and minima are those of the differential calculus or, in geometrical
problems, what is practically Fermat’s method. Some
problems may be solved by algebra; thus if *y* = ƒ(*x*) ÷ φ(*x*),
where ƒ(*x*) and φ(*x*) are polynomials in *x*, the limits to the
values of *y*φ may be found from the consideration that the
equation *y*φ(*x*) − ƒ(*x*) = 0 must have real roots. This is a
useful method in the case in which φ(*x*) and ƒ(*x*) are quadratics,
but scarcely ever in any other case. The problem of
finding the maximum product of *n* positive quantities whose
sum is given may also be found, algebraically, thus. If *a* and *b*
are any two real unequal quantities whatever {12(*a* + *b*)}^{2} > *ab*,
so that we can increase the product leaving the sum unaltered
by replacing any two terms by half their sum, and
so long as any two of the quantities are unequal we can increase
the product. Now, the quantities being all positive, the product
cannot be increased without limit and must somewhere attain a
maximum, and no other form of the product than that in which
they are all equal can be the maximum, so that the product is
a maximum when they are all equal. Its minimum value
is obviously zero. If the restriction that all the quantities
shall be positive is removed, the product can be made equal
to any quantity, positive or negative. So other theorems
of algebra, which are stated as theorems on inequalities, may
be regarded as algebraic solutions of problems on maxima and
minima.

For purely geometrical questions the only general method
available is practically that employed by Fermat. If a quantity
depends on the position of some point P on a curve, and if its
value is equal at two neighbouring points P and P′, then at some
position between P and P′ it attains a maximum or minimum, and
this position may be found by making P and P′ approach each
other indefinitely. Take for instance the problem of Regiomontanus
“to find a point on a given straight line which subtends
a maximum angle at two given points A and B.” Let P and P′
be two near points on the given straight line such that the angles
APB and AP′B are equal. Then ABPP′ lie on a circle. By
making P and P′ approach each other we see that for a maximum
or minimum value of the angle APB, P is a point in which a circle
drawn through AB touches the given straight line. There are
two such points, and unless the given straight line is at right
angles to AB the two angles obtained are not the same. It is
easily seen that both angles are maxima, one for points on the
given straight line on one side of its intersection with AB, the
other for points on the other side. For further examples of this
method together with most other geometrical problems on
maxima and minima of any interest or importance the reader may
consult such a book as J. W. Russell’s *A Sequel to Elementary*
*Geometry* (Oxford, 1907).

The method of the differential calculus is theoretically very
simple. Let *u* be a function of several variables *x*_{1}, *x*_{2}, *x*_{3} . . . *x*_{n},
supposed for the present independent; if *u* is a maximum or
minimum for the set of values *x*_{1}, *x*_{2}, *x*_{3}, . . . *x*_{n}, and *u* becomes
*u* + δ*u*, when *x*_{1}, *x*_{2}, *x*_{3} . . . *x*_{n} receive small increments δ*x*_{1},
δ*x*_{2}, . . . δ*x*_{n}; then δ*u* must have the same sign for all possible
values of δ*x*_{1}, δ_{2} . . . δ*x*_{n}.

Now

δu = Σ | δu |
δx_{1} + 12 { Σ | δ^{2}u |
δx_{1}^{2} + 2Σ | δ^{2}u |
δx_{1}δx_{2} . . . } + . . . . |

δx_{1} | δx_{1}^{2} |
δx_{1}δx_{2} |

The sign of this expression in general is that of Σ(δ*u*/δ*x*_{1})δ*x*_{1},
which cannot be one-signed when *x*_{1}, *x*_{2}, . . . *x*_{n} can take all
possible values, for a set of increments δ*x*_{1}, δ*x*_{2} . . . δ*x*_{n}, will give an
opposite sign to the set −δ*x*_{1}, −δ*x*_{2}, . . . −δ*x*_{n}. Hence Σ(δ*u*/δ*x*_{1})δ*x*_{1}
must vanish for all sets of increments δ*x*_{1}, . . . δ*x*_{n}, and since
these are independent, we must have δ*u*/δ*x*_{1} = 0, δ*u*/δ*x*_{2} = 0, . . .
δ*u*/δ*x*_{n} = 0. A value of *u* given by a set of solutions of these equations
is called a “critical value” of *u*. The value of δ*u* now becomes

12 { Σ | δ^{2}u |
δx_{1}^{2} + 2 Σ | δ^{2}u |
δx_{1}δx_{2} + . . . }; |

δx_{1}^{2} | δx_{1}δx_{2} |

for *u* to be a maximum or minimum this must have always the same
sign. For the case of a single variable *x*, corresponding to a value
of *x* given by the equation *du*/*dx* = 0, *u* is a maximum or minimum
as *d* ^{2}*u*/*dx*^{2} is negative or positive. If *d* ^{2}*u*/*dx*^{2} vanishes, then there
is no maximum or minimum unless *d* ^{2}*u*/*dx*^{2} vanishes, and there is
a maximum or minimum according as *d* ^{4}*u*/*dx*^{4} is negative or positive.
Generally, if the first differential coefficient which does not vanish
is even, there is a maximum or minimum according as this is negative
or positive. If it is odd, there is no maximum or minimum.

In the case of several variables, the quadratic

Σ | δ^{2}u |
δx_{1}^{2} + 2 Σ | δ^{2}u |
δx_{1}δx_{2} + . . . |

δx_{1}^{2} | δx_{1}δx_{2} |

must be one-signed. The condition for this is that the series of discriminants

a_{11} , | a_{11} a_{12} | , | a_{11} a_{12} a_{13} | , . . . |

a_{21} a_{22} | a_{21} a_{22} a_{23} | |||

a_{31} a_{32} a_{33} |

where *a*_{pq} denotes δ^{2}*u*/δ*a*_{p}δ*a*_{q} should be all positive, if the quadratic
is always positive, and alternately negative and positive, if the
quadratic is always negative. If the first condition is satisfied the
critical value is a minimum, if the second it is a maximum. For
the case of two variables the conditions are

δ^{2}u |
· | δ^{2}u |
> ( | δ^{2}u |
)2 |

δx_{1}^{2} | δx_{2}^{2} |
δx_{1}δx_{2} |

for a maximum or minimum at all and δ^{2}*u*/δ*x*_{1}^{2} and δ^{2}*u*/δ*x*_{2}^{2} both
negative for a maximum, and both positive for a minimum. It is
important to notice that by the quadratic being one-signed is meant
that it cannot be made to vanish except when δ*x*_{1}, δ*x*_{2}, . . . δ*x*_{n} all
vanish. If, in the case of two variables,

δ^{2}u |
· | δ^{2}u |
= ( | δ^{2}u |
)2 |

δx_{1}^{2} | δx_{2}^{2} |
δx_{1}δx_{2} |

then the quadratic is one-signed unless it vanishes, but the value
of *u* is not necessarily a maximum or minimum, and the terms of
the third and possibly fourth order must be taken account of.

Take for instance the function *u* = *x*^{2} − *xy*^{2} + *y*^{2}. Here the values
*x* = 0, *y* = 0 satisfy the equations δ*u*/δ*x* = 0, δ*u*/δ*y* = 0, so that zero
is a critical value of *u*, but it is neither a maximum nor a minimum
although the terms of the second order are (δ*x*)^{2}, and are never
negative. Here δ*u* = δ*x*^{2} − δ*x*δ*y*^{2} + δ*y*^{2}, and by putting δ*x* = 0 or an
infinitesimal of the same order as δ*y*^{2}, we can make the sign of δ*u*
depend on that of δ*y*^{2}, and so be positive or negative as we please.
On the other hand, if we take the function *u* = *x*^{2} − *xy*^{2} + *y*^{4}, *x* = 0, *y* = 0
make zero a critical value of *u*, and here δ*u* = δ*x*^{2} − δ*x*δ*y*^{2} + δ*y*^{4}, which
is always positive, because we can write it as the sum of two squares,
viz. (δ*x* − 12δ*y*^{2})^{2} + 34δ*y*^{4}; so that in this case zero is a minimum value
of *u*.

A critical value usually gives a maximum or minimum in the
case of a function of one variable, and often in the case of several
independent variables, but all maxima and minima, particularly
absolutely greatest and least values, are not necessarily critical
values. If, for example, *x* is restricted to lie between the values
*a* and *b* and φ′(*x*) = 0 has no roots in this interval, it follows that
φ′(*x*) is one-signed as *x* increases from *a* to *b*, so that φ(*x*) is increasing
or diminishing all the time, and the greatest and least values of
φ(*x*) are φ(*a*) and φ(*b*), though neither of them is a critical value.
Consider the following example: A person in a boat a miles from
the nearest point of the beach wishes to reach as quickly as possible
a point *b* miles from that point along the shore. The ratio of his
rate of walking to his rate of rowing is cosec α. Where should
he land?

Here let AB be the direction of the beach, A the nearest point
to the boat O, and B the point he wishes to reach. Clearly he
must land, if at all, between A and B. Suppose he lands at P.
Let the angle AOP be θ, so that OP = *a* secθ, and PB = *b* − *a* tan θ.
If his rate of rowing is V miles an hour his time will be a sec θ/V +
(*b* − *a* tan θ) sin α/V hours. Call this T. Then to the first power
of δθ, δT = (*a*/V) sec^{2}θ (sin θ − sin α)δθ, so that if AOB > α, δT and δθ
have opposite signs from θ = 0 to θ = α, and the same signs from
θ = α to θ = AOB. So that when AOB is > α, T decreases from θ = 0
to θ = α, and then increases, so that he should land at a point distant
a tan α from A, unless *a* tan α > *b*. When this is the case, δT and δθ
have opposite signs throughout the whole range of θ, so that T
decreases as θ increases, and he should row direct to B. In the
first case the minimum value of T is also a critical value; in the second
case it is not.

The greatest and least values of the bending moments of loaded rods are often at the extremities of the divisions of the rods and not at points given by critical values.

In the case of a function of several variables, X_{1}, *x*_{2}, . . . *x*_{n},
not independent but connected by *m* functional relations *u*_{1} = 0,
*u*_{2} = 0, . . ., *u*_{m} = 0, we might proceed to eliminate *m* of the
variables; but Lagrange’s “Method of undetermined Multipliers”
is more elegant and generally more useful.

We have δ*u*_{1} = 0, δ*u*_{2} = 0, . . ., δ*u*_{m} = 0. Consider instead of
δ*u*, what is the same thing, viz., δ*u* + λ_{1}δ*u*_{1} + λ_{2}δ*u*_{2} + . . . + λ_{m} δ*u*_{m},
where λ_{1}, λ_{2}, . . . λ_{m}, are arbitrary multipliers. The terms of the
first order in this expression are

Σ | δu |
δx_{1} + λ_{1} Σ | δu_{1} |
δx_{1} + . . . + λ_{m} Σ | δu_{m} |
δx_{1}. |

δx_{1} | δx_{1} |
δx_{1} |

We can choose λ_{1}, . . . λ_{m}, to make the coefficients of δ*x*_{1}, δ*x*_{2},
... δ*x*_{m}, vanish, and the remaining δ*x*_{m+1} to δ*x*_{n} may be regarded
as independent, so that, when *u* has a critical value, their coefficients
must also vanish. So that we put

δu |
+ | δu_{1} |
+ . . . + λ_{m} | δu_{m} |
= 0 |

δx_{r} | δx_{r} |
δx_{r} |

for all values of *r*. These equations with the equations *u*_{1} = 0, . . .,
*u*_{m} = 0 are exactly enough to determine λ_{1}, . . ., λ_{m}, *x*_{1} *x*_{2}, . . ., *x*_{n},
so that we find critical values of *u*, and examine the terms of the
second order to decide whether we obtain a maximum or minimum.

To take a very simple illustration; consider the problem of determining
the maximum and minimum radii vectors of the ellipsoid
*x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} + *z*^{2}/*c*^{2} = 1, where *a*^{2} > *b*^{2} > *c*^{2}. Here we require the maximum
and minimum values of *x*^{2} + *y*^{2} + *z*^{2} where *x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} + *z*^{2}/*c*^{2} = 1.

We have

δu = 2xδx ( 1 + | λ | ) + 2yδy ( | λ | ) + 2zδz ( | λ | ) |

a^{2} | b^{2} |
c^{2} |

+ δx^{2} ( 1 + | λ | ) + δy^{2} ( | λ | ) + δz^{2} ( | λ | ). |

a^{2} | b^{2} |
c^{2} |

To make the terms of the first order disappear, we have the three equations:—

*x*(1 + λ/

*a*

^{2}) = 0,

*y*(1 + λ/

*b*

^{2}) = 0,

*z*(1 + λ/

*c*

^{2}) = 0.

*x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} + *z*^{2}/*c*^{2} = 1, *a*^{2} > *b*^{2} > *c*^{2}, viz.:—

(1) y = 0, z = 0, λ = −a^{2}; (2) z = 0, x = 0, λ = −b^{2}; |

(3) x = 0, y = 0, λ = −c^{2}. |

In the case of (1) δ*u* = δ*y*^{2} (1 − *a*^{2}/*b*^{2}) + δ*z*^{2} (1 − *a*^{2}/*c*^{2}), which is
always negative, so that *u* = *a*^{2} gives a maximum.

In the case of (3) δ*u* = δ*x*^{2} (1 − *c*^{2}/*a*^{2}) + δ*y*^{2} (1 − *c*^{2}/*b*^{2}), which is
always positive, so that *u* = *c*^{2} gives a minimum.

In the case of (2) δ*u* = δ*x*^{2} (1 − *b*^{2}/*a*^{2}) − δ*z*^{2}(*b*^{2}/*c*^{2} − 1), which can be
made either positive or negative, or even zero if we move in the
planes *x*^{2} (1 − *b*^{2}/*a*^{2}) = *z*^{2} (*b*^{2}/*c*^{2} − 1), which are well known to be the
central planes of circular section. So that *u* = *b*^{2}, though a critical
value, is neither a maximum nor minimum, and the central planes
of circular section divide the ellipsoid into four portions in two of
which *a*^{2} > *r* ^{2} > *b*^{2}, and in the other two *b*^{2} > *r* ^{2} > *c*^{2}. (A. E. J.)