y) = 3x2 + 4xy + y2 → extr, x + y = 1.46 46) f(x, y, z) = xy2z3 → extr, x + y + z = 1.
47 47) f(x, y, z) = xyz → extr, x2 + y2 + z2 = 1, x + y + z = 0.
48 48) f(x, y, z) = a2x2 + b2y2 + c2z2 − (ax2 + by2 + cz2)2 → extr, x2 + y2 + z2 = 1, a > b > c > 0.
49 49) f(x, y, z) = x + y + z2 + 2(xy + yz + zx) → extr,x2 + y2 + z = 1.
50 50) f(x, y, z) = x − 2y + 2z → extr, x2 + y2 + z2 = 1.
51 51) f(x, y, z) = xm yn zp → extr, x + y + z = a, m > 0, n > 0, p > 0, a > 0.
52 52) f(x, y, z) = x2 + y2 + z2 → extr, x2/a2 + y2/b2 + z2/c2 = 1, a > b > c > 0.
53 53) f(x, y, z) = xy2z3 → extr, x + 2y + 3z = a, x > 0, y > 0, z > 0, a > 0.
54 54) f(x, y, z) = xy + yz → extr, x2 + y2 = 2, y + z = 2, x > 0, y > 0, z > 0.
55 55) f(x, y, z) = sin (x) sin (y) sin(z) → extr, x + y + z = π/2.
56 56) f(x, y) = ex−y − x − y → extr, x + y ≤ 1, x ≥ 0, y ≥ 0.
57 57) f(x, y) = x2 + y2 − 2x − 4y → extr, 2x + 3y − 6 ≤ 0, x + 4y − 5 ≤ 0.
58 58) f(x, y) = 2xy − x2 − 2y2 → extr, x − y + 1 ≥ 0, 2x + 3y + 6 ≤ 0.
59 59) f(x, y) = x2 + y2 → extr, −5x + 4y ≤ 0, −x + 4y + 3 ≤ 0.
60 60) f(x, y) = x2 + y2 − 2x → extr, x − 2y + 2 ≤ 0, 2x − y ≥ 0.
61 61) f(x, y, z) = xyz → extr, x2 + y2 + z2 ≤ 1.
62 62) f(x, y, z) = 2x2 + 2x + 4y −3z → extr, 8x −3y + 3z ≤ 40,− 2x + y −z = −3, y ≥ 0.
63 63) f(x, y, z) = x2 + 4y2 + z2 → extr, x + y + z ≤ 12, x ≥ 0, y ≥ 0, z ≥ 0.
64 64) f(x, y, z) = 3y2 − 11x − 3y −z → extr, x − 7y + 3z + 7 ≤ 0, 5x + 2y −z ≤ 2, z ≥ 0.
65 65) f(x, y, z) = xz − 2y → extr, 2x − y − 3z ≤ 10, 3x + 2y + z = 6, y ≥ 0.
66 66) f(x, y, z) = −4x − y + z2 → extr, x2 + y2 + xz − 1 ≤ 0, x2 + y2 − 2y ≤ 0, 5 − x + y + z ≤ 0, x ≥ 0, y ≥ 0, z ≥ 0.
67 67) , , b > 0, xj ≥ 0, αj > 0, βj > 0, aj > 0, j = 1, 2, … , n.
68 68) , , b > 0, xj ≥ 0, cj > 0, αj > 0, βj > 0, j = 1, 2, … , n.
69 69) , , b > 0, xj > 0, cj > 0, αj > 0, βj > 0, j = 1, 2, … , n.
70 70) , b > 0, α > 0, xj > 0, cj > 0, j = 1, 2, … , n.
71 71) , , b > 0, 0 < α < 1, xj > 0, cj > 0, j = 1, 2, … , n.
72 72) , , cj > 0, a = (a1, … , an) ≠ 0, α = 2m, m ∈ N.
73 73) , , cj > 0, a = (a1, … , an) ≠ 0, α > 1, b > 0.
74 74) , , b > 0, aj > 0, c = (c1, … , cn) ≠ 0, α = 2m, m ∈ N.
75 75) , , b > 0, aj > 0, c = (c1, … , cn) ≠ 0, α > 1.
76 76) , , b > 0, c = (c1, … , cn) ≠ 0, α > 1.
77 77) Divide the number 8 into two parts so that the product of their product on the difference is maximal (Niccolo Tartaglia problem).
78 78) Determine the rectangular triangle of the largest area, provided that the sum of the lengths of its legs is equal to a given number (Fermat’s problem).
79 79) On the BC side of a triangle ABC, define a point E such that the parallelogram ADEK, whose points D and K lie on sides AB and AC, respectively, has the largest area (Euclid problem).
80 80) On a given face of a tetrahedron, take a point through which planes parallel to three other faces are drawn. Choose a point so that the volume of the parallelepiped is maximal (generalized Euclid problem).
81 81) Determine a polynomial of the second-degree t2 + x1t + x2 such that the integralgets the smallest value (Legendre problem for polynomial of the second degree).
82 82) Determine a polynomial of the third degree t3 + x1t2 + x2t + x3 such that the integralgets the smallest value (Legendre problem for polynomial of the third degree).
83 83) Among all discrete random variables that take n values, determine a random variable with the largest entropy. The entropy of the sequence of positive numbers p1, … , pn, such that , is the number
84 84) Insert a rectangle of maximum area into a circle.
85 85) Insert a triangle of maximum area into a circle.
86 86) Insert a cylinder with maximum volume into a ball (Kepler’s problem).
87 87) Insert a cone with maximum volume into a ball.
88 88) Among cones inscribed in a ball, determine a cone with the maximum area of the lateral surface.
89 89) Insert in a sphere from the space ℝn a rectangular parallelepiped with the largest volume.
90 90) Insert a tetrahedron with the largest volume into a ball.
91 91) Among triangles with a given perimeter determine a triangle of the largest area.
92 92) Among all n-angles of a given perimeter determine an n-cube of the largest area (Zeno’s problem).
93 93) Insert n-angles of the maximum area in a circle.
94 94) On the diameter AB of a circle of the unit radius, a point E is taken through which a chord CD is drawn. Determine a position of the chord in which the square of the quadrilateral ABCD is maximal.
95 95) Determine in a triangle such a point that the sum of the ratio of lengths of sides and distances from the point to relevant sides is minimal.
96 96) Insert into a circle a triangle with the largest sum of squares of sides.
97 97) Through a given point inside a corner, draw a segment with ends on the sides of the corner so that the area of the formed triangle is minimal.
98 98) Through a point inside a corner draw a section with ends on the sides of the corner so that the perimeter of the formed triangle is minimal.
99 99) Determine a quadrilateral with given sides of the largest area.
100 100) Among segments of a ball having a given area of the lateral surface, find the segment with the largest volume (Archimedes’ problem).
101 101) Determine a point C on a line such that the sum of the distances from the point C to the given points A and B is minimal.
102 102) Among all tetrahedra with a given base and height, find a tetrahedron with the smallest lateral surface.
103 103) Among
