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. , ik of k indices, each assuming the values 1 and 2,is meant).Example 2 If f (x) = f (x 1 , . . . , x m ) and f ∈ C (k) (G; R), then, under the assumption that [x, x + h] ⊂ G, for the function ϕ(t) = f (x + th) defined on the closedinterval [0, 1] we obtainϕ (k) (t) = hi1 · · · hik ∂i1 ···ik f (x + th),(8.58)where summation over all sets of indices i1 , . . . , ik , each assuming all values from1 to m inclusive, is meant on the right.We can also write formula (8.58) askϕ (k) (t) = h1 ∂1 + · · · + hm ∂m f (x + th).(8.59)8.4 Real-valued Functions of Several Variables4618.4.4 Taylor’s FormulaTheorem 4 If the function f : U (x) → R is defined and belongs to class C (n) (U (x);R) in a neighborhood U (x) ⊂ Rm of the point x ∈ Rm , and the closed interval[x, x + h] is completely contained in U (x), then the following equality holds:f x 1 + h1 , . .
. , x m + hm − f x 1 , . . . , x m ==n−1k1 1h ∂1 + · · · + hm ∂m f (x) + rn−1 (x; h),k!(8.60)k=1wheren(1 − t)n−1 1h ∂1 + · · · + hm ∂m f (x + th) dt.(n − 1)!1rn−1 (x; h) =0(8.61)Equality (8.60), together with (8.61), is called Taylor’s formula with integralform of the remainder.Proof Taylor’s formula follows immediately from the corresponding Taylor formulafor a function of one variable. In fact, consider the auxiliary functionϕ(t) = f (x + th),which, by the hypotheses of Theorem 4, is defined on the closed interval 0 ≤ t ≤ 1and (as we have verified above) belongs to the class C (n) [0, 1].Then for τ ∈ [0, 1], by Taylor’s formula for functions of one variable, we canwrite that11 ϕ (0)τ + · · · +ϕ (n−1) (0)τ n−1 +1!(n − 1)!ϕ(τ ) = ϕ(0) ++10(1 − t)n−1 (n)ϕ (tτ )τ n dt.(n − 1)!Setting τ = 1 here, we obtainϕ(1) = ϕ(0) ++0111 ϕ (0) + · · · +ϕ (n−1) (0) +1!(n − 1)!(1 − t)n−1 (n)ϕ (t) dt.(n − 1)!(8.62)Substituting the valueskϕ (k) (0) = h1 ∂1 + · · · + hm ∂m f (x)(k = 0, .
. . , n − 1),4628 Differential Calculus in Several Variablesnϕ (n) (t) = h1 ∂1 + · · · + hm ∂m f (x + th),into this equality in accordance with formula (8.59), we find what Theorem 4 asserts.Remark If we write the remainder term in relation (8.62) in the Lagrange formrather than the integral form, then the equalityϕ(1) = ϕ(0) +1 11ϕ (0) + · · · +ϕ (n−1) (0) + ϕ (n) (θ ),1!(n − 1)!n!where 0 < θ < 1, implies Taylor’s formula (8.60) with remainder termrn−1 (x; h) =n1 1h ∂1 + · · · + hm ∂m f (x + θ h).n!(8.63)This form of the remainder term, as in the case of functions of one variable, iscalled the Lagrange form of the remainder term in Taylor’s formula.Since f ∈ C (n) (U (x); R), it follows from (8.63) thatrn−1 (x; h) =n1 1h ∂1 + · · · + hm ∂m f (x) + o ,h,nn!as h → 0,and so we have the equalityf x 1 + h1 , . . .
, x m + hm − f x 1 , . . . , x m ==nk1 1h ∂1 + · · · + hm ∂m f (x) + o ,h,nk!as h → 0,(8.64)k=1called Taylor’s formula with the remainder term in Peano form.8.4.5 Extrema of Functions of Several VariablesOne of the most important applications of differential calculus is its use in findingextrema of functions.Definition 1 A function f : E → R defined on a set E ⊂ Rm has a local maximum(resp. local minimum) at an interior point x0 of E if there exists a neighborhoodU (x0 ) ⊂ E of the point x0 such that f (x) ≤ f (x0 ) (resp. f (x) ≥ f (x0 )) for allx ∈ U (x0 ).If the strict inequality f (x) < f (x0 ) holds for x ∈ U (x0 ) \ x0 (or, respectively,f (x) > f (x0 )), the function has a strict local maximum (resp. strict local minimum)at x0 .8.4 Real-valued Functions of Several Variables463Definition 2 The local minima and maxima of a function are called its local extrema.Theorem 5 Suppose a function f : U (x0 ) → R defined in a neighborhood U (x0 ) ⊂Rm of the point x0 = (x01 , .
. . , x0m ) has partial derivatives with respect to each of thevariables x 1 , . . . , x m at the point x0 .Then a necessary condition for the function to have a local extremum at x0 is thatthe following equalities hold at that point:∂f(x0 ) = 0,∂x 1...,∂f(x0 ) = 0.∂x m(8.65)Proof Consider the function ϕ(x 1 ) = f (x 1 , x02 , . . . , x0m ) of one variable defined,according to the hypotheses of the theorem, in some neighborhood of the point x01on the real line. At x01 the function ϕ(x 1 ) has a local extremum, and since ∂f ϕ x01 = 1 x01 , x02 , . . .
, x0m ,∂x∂fit follows that ∂x1 (x0 ) = 0.The other equalities in (8.65) are proved similarly.We call attention to the fact that relations (8.65) give only necessary but notsufficient conditions for an extremum of a function of several variables. An examplethat confirms this is any example constructed for this purpose for functions of onevariable. Thus, where previously we spoke of the function x → x 3 , whose derivativeis zero at zero, but has no extremum there, we can now consider the function 3f x1, . . .
, xm = x1 ,all of whose partial derivatives are zero at x0 = (0, . . . , 0), while the function obviously has no extremum at that point.Theorem 5 shows that if the function f : G → R is defined on an open setG ⊂ Rm , its local extrema are found either among the points at which f is notdifferentiable or at the points where the differential df (x0 ) or, what is the same, thetangent mapping f (x0 ), vanishes.We know that if a mapping f : U (x0 ) → Rn defined in a neighborhood U (x0 ) ⊂mR of the point x0 ∈ Rm is differentiable at x0 , then the matrix of the tangent mapping f (x0 ) : Rm → Rn has the form⎛∂ 1 f 1 (x0 )..⎜⎝.∂1f n (x0)···...···⎞∂m f 1 (x0 )..⎟⎠..∂mf n (x0)(8.66)4648 Differential Calculus in Several VariablesDefinition 3 The point x0 is a critical point of the mapping f : U (x0 ) → Rn if therank of the Jacobi matrix (8.66) of the mapping at that point is less than min{m, n},that is, smaller than the maximum possible value it can have.In particular, if n = 1, the point x0 is critical if condition (8.65) holds, that is, allthe partial derivatives of the function f : U (x0 ) → R vanish.The critical points of real-valued functions are also called the stationary pointsof these functions.After the critical points of a function have been found by solving the system(8.65), the subsequent analysis to determine whether they are extrema or not canoften be carried out using Taylor’s formula and the following sufficient conditionsfor the presence or absence of an extremum provided by that formula.Theorem 6 Let f : U (x0 ) → R be a function of class C (2) (U (x0 ); R) defined in aneighborhood U (x0 ) ⊂ Rm of the point x0 = (x01 , .
. . , x0m ) ∈ Rm , and let x0 be acritical point of the function f .If, in the Taylor expansion of the function at the point x0f x01 + h1 , . . . , x0m + hm =m 1 ∂ 2f= f x01 , . . . , x0m +(x0 )hi hj + o ,h,2ij2!∂x ∂x(8.67)i,j =1the quadratic formmi,j =1∂ 2f(x0 )hi hj ≡ ∂ij f (x0 )hi hj∂x i ∂x j(8.68)a) is positive-definite or negative-definite, then the point x0 has a local extremumat x0 , which is a strict local minimum if the quadratic form (8.68) is positive-definiteand a strict local maximum if it is negative-definite;b) assumes both positive and negative values, then the function does not have anextremum at x0 .Proof Let h = 0 and x0 + h ∈ U (x0 ).
Let us represent (8.67) in the form?> mi hj ∂ 2f1hf (x0 + h) − f (x0 ) = ,h,2+ o(1) ,(x0 )2!∂x i ∂x j,h, ,h,(8.69)i,j =1where o(1) is infinitesimal as h → 0.It is clear from (8.69) that the sign of the difference f (x0 + h) − f (x0 ) is completely determined by the sign of the quantity in brackets. We now undertake tostudy this quantity.The vector e = (h1 /,h,, . . . , hm /,h,) obviously has norm 1. The quadratic form(8.68) is continuous as a function h ∈ Rm , and therefore its restriction to the unit8.4 Real-valued Functions of Several Variables465sphere S(0; 1) = {x ∈ Rm |,x, = 1} is also continuous on S(0; 1). But the sphere Sis a closed bounded subset in Rm , that is, it is compact.
Consequently, the form(8.68) has both a minimum point and a maximum point on S, at which it assumesrespectively the values m and M.If the form (8.68) is positive-definite, then 0 < m ≤ M, and there is a numberδ > 0 such that |o(1)| < m for ,h, < δ. Then for ,h, < δ the bracket on the righthand side of (8.69) is positive, and consequently f (x0 + h) − f (x0 ) > 0 for 0 <,h, < δ. Thus, in this case the point x0 is a strict local minimum of the function.One can verify similarly that when the form (8.68) is negative-definite, the function has a strict local maximum at the point x0 .Thus a) is now proved.We now prove b).Let em and eM be points of the unit sphere at which the form (8.68) assumes thevalues m and M respectively, and let m < 0 < M.Setting h = tem , where t is a sufficiently small positive number (so small thatx0 + tem ∈ U (x0 )), we find by (8.69) thatf (x0 + tem ) − f (x0 ) =1 2t m + o(1) ,2!where o(1) → 0 as t → 0.
Starting at some time (that is, for all sufficiently smallvalues of t), the quantity m + o(1) on the right-hand side of this equality will havethe sign of m, that is, it will be negative. Consequently, the left-hand side will alsobe negative.Similarly, setting h = teM , we obtainf (x0 + teM ) − f (x0 ) =1 2t M + o(1) ,2!and consequently for all sufficiently small t the difference f (x0 + teM ) − f (x0 ) ispositive.Thus, if the quadratic form (8.68) assumes both positive and negative values onthe unit sphere, or, what is obviously equivalent, in Rm , then in any neighborhoodof the point x0 there are both points where the value of the function is larger thanf (x0 ) and points where the value is smaller than f (x0 ).
Hence, in that case x0 isnot a local extremum of the function.We now make a number of remarks in connection with this theorem.Remark 1 Theorem 6 says nothing about the case when the form (8.68) is semidefinite, that is, nonpositive or nonnegative. It turns out that in this case the pointmay be an extremum, or it may not. This can be seen, in particular from the following example.Example 3 Let us find the extrema of the function f (x, y) = x 4 + y 4 − 2x 2 , whichis defined in R2 .4668 Differential Calculus in Several VariablesIn accordance with the necessary conditions (8.65) we write the system of equations⎧∂f3⎪⎪⎨ ∂x (x, y) = 4x − 4x = 0,⎪ ∂f⎪⎩ (x, y) = 4y 3 = 0,∂yfrom which we find three critical points: (−1, 0), (0, 0), (1, 0).Since∂ 2f(x, y) = 12x 2 − 4,∂x 2∂ 2f(x, y) ≡ 0,∂x∂y∂ 2f(x, y) = 12y 2 ,∂y 2at the three critical points the quadratic form (8.68) has respectively the form 28 h1 , 2−4 h1 , 28 h1 .That is, in all cases it is positive semi-definite or negative semi-definite.
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