For any positive real numbers a and B, the size relation between (a ^ 2 + B ^ 2) / 2 and ab is studied For any positive real number a, B, the size relation with AB is studied (1) Substituting the numerical value, comparing the size, finding the law ① When a = 3, B = 1, (a ^ 2 + B ^ 2) / 2 > AB; ② When a = root 3, B = root 3, (a ^ 2 + B ^ 2) / 2___ ab; ③ a=___ ,b=___ (a ^ 2 + B ^ 2) / 2___ ab; Conjecture: for any positive real number a, B, (a ^ 2 + B ^ 2) / 2___ ab. (2) Construct graph to verify conjecture The algebraic formula (a ^ 2 + B ^ 2) / 2 can be expressed by the sum of the areas of two isosceles right triangles whose waist length is a and B respectively. The above conjecture can be verified by means of the splicing and segmentation of the two triangles (3) Application Explore: the maximum area of a right triangle with a hypotenuse of 5

For any positive real numbers a and B, the size relation between (a ^ 2 + B ^ 2) / 2 and ab is studied For any positive real number a, B, the size relation with AB is studied (1) Substituting the numerical value, comparing the size, finding the law ① When a = 3, B = 1, (a ^ 2 + B ^ 2) / 2 > AB; ② When a = root 3, B = root 3, (a ^ 2 + B ^ 2) / 2___ ab; ③ a=___ ,b=___ (a ^ 2 + B ^ 2) / 2___ ab; Conjecture: for any positive real number a, B, (a ^ 2 + B ^ 2) / 2___ ab. (2) Construct graph to verify conjecture The algebraic formula (a ^ 2 + B ^ 2) / 2 can be expressed by the sum of the areas of two isosceles right triangles whose waist length is a and B respectively. The above conjecture can be verified by means of the splicing and segmentation of the two triangles (3) Application Explore: the maximum area of a right triangle with a hypotenuse of 5

Answer: (Party A + Party B) / 2) AB proves that (Party A + Party B) 2Ab is obtained by multiplying 2 on both sides of the above formula, and (Party A + Party B) - 2Ab 0 is obtained by shifting the term, that is (Party a-b) 0. Because the last inequality holds, the original proposition holds