Final answer:
The statement is true. When two functions f and g are both increasing, their product fg is also increasing; this follows from the properties of multiplication and the definition of increasing functions.
Step-by-step explanation:
To determine whether the statement If f and g are increasing, the product fg is increasing is true or false, we need to understand the behavior of the product of two functions. To say a function is increasing means that if x1 < x2 then f(x1) < f(x2) for any x1 and x2 in the domain of the function.
Let's consider two functions f and g that are both increasing. Now suppose we take any two points, say x1 and x2, where x1 < x2. Since f is increasing, f(x1) < f(x2). Similarly, for g, we have g(x1) < g(x2). To see if their product is also increasing, we compare fg(x1) with fg(x2). We have fg(x1) = f(x1)g(x1) and fg(x2) = f(x2)g(x2). Since both f(x1) and g(x1) are less than f(x2) and g(x2) respectively, and because the multiplication of positive number is commutative and associative, it follows that fg(x1) < fg(x2). Hence, the product fg is also increasing if both f and g are increasing.