5. if two numbers “a” and “b” have a sum of 10, then will the product of the two numbers have a maximum or a…

Explanation:

Step1: Express one variable in terms of the other

Given (a + b=10), we can express (b = 10 - a).

Step2: Form the product function

The product (P=a\times b=a(10 - a)=10a - a^{2}). This is a quadratic function of the form (y = Ax^{2}+Bx + C), where (A=- 1), (B = 10), (C = 0).

Step3: Analyze the quadratic function

For a quadratic function (y = Ax^{2}+Bx + C), if (A<0), the parabola opens downwards. Since (A=-1<0) in (P =-a^{2}+10a), the function has a maximum value. The vertex of the parabola (y = Ax^{2}+Bx + C) has its (x) - coordinate (in our case, the value of (a) for which (P) is extremum) given by (x=-\frac{B}{2A}). Here, (a=-\frac{10}{2\times(-1)} = 5), and (P(5)=10\times5-5^{2}=25).

Answer:

The product of the two numbers has a maximum.