Question

1. if two numbers “a” and “b” have a difference of 10, then will the product of the two numbers have a maximum or a minimum? explain. justify your answer

Explanation:

Step1: Express one variable in terms of the other

Given \(a - b=10\), so \(a = b + 10\). The product \(P=ab=(b + 10)b=b^{2}+10b\).

Step2: Analyze the quadratic function

The function \(P(b)=b^{2}+10b\) is a quadratic function of the form \(y = Ax^{2}+Bx + C\), where \(A = 1\), \(B=10\), \(C = 0\).
Since \(A=1>0\), the parabola \(y = P(b)\) opens upwards.

Step3: Use the vertex formula

The vertex of a quadratic function \(y=Ax^{2}+Bx + C\) has its \(x\) - coordinate at \(x=-\frac{B}{2A}\). For \(P(b)=b^{2}+10b\), the \(b\) - value of the vertex is \(b=-\frac{10}{2\times1}=- 5\).
The minimum value of \(P(b)\) occurs at \(b=-5\), and \(P(-5)=(-5)^{2}+10\times(-5)=25 - 50=-25\).

Answer:

The product of the two numbers has a minimum.