Question

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

Explanation:

Step1: Express \(b\) in terms of \(a\)

Since \(a + b=10\), then \(b = 10 - a\).

Step2: Define the product function

Let \(P(a)\) be the product of \(a\) and \(b\). So \(P(a)=a\times b=a(10 - a)=10a - a^{2}\).

Step3: Analyze the function type

The function \(P(a)=-a^{2}+10a\) is a quadratic function. For a quadratic function \(y = Ax^{2}+Bx + C\) (\(A
eq0\)), here \(A=- 1\), \(B = 10\), \(C = 0\).

Step4: Use the property of quadratic functions

The graph of a quadratic function \(y = Ax^{2}+Bx + C\) is a parabola. If \(A<0\), the parabola opens downwards. The vertex of the parabola \(y = Ax^{2}+Bx + C\) has the \(x\) - coordinate \(x=-\frac{B}{2A}\). For \(P(a)=-a^{2}+10a\), \(A=-1\), \(B = 10\). The \(a\) - value of the vertex is \(a=-\frac{10}{2\times(-1)} = 5\). And \(P(5)=5\times(10 - 5)=25\). Since the parabola \(y=-a^{2}+10a\) opens downwards (because \(A=-1<0\)), the function \(P(a)\) has a maximum value.

Answer:

The product of the two numbers has a maximum.