give an example of two decreasing functions whose product is increasing
-(e^x + 1)
-arctan(x)
To find an example of two decreasing functions whose product is increasing, let's first understand the concept.
For the product of two functions to be increasing, it means that as the independent variable (x) increases, the value of the product should also increase. On the other hand, for a function to be decreasing, it means that as x increases, the value of the function decreases.
To create a specific example, let's consider two simple functions: f(x) = -x and g(x) = -1/x.
First, let's look at f(x) = -x. As x increases, f(x) decreases. For example, when x = 1, f(x) = -1. But when x = 2, f(x) = -2, which is smaller than -1.
Next, let's examine g(x) = -1/x. Again, as x increases, g(x) decreases. For instance, when x = 1, g(x) = -1. However, when x = 2, g(x) = -1/2, which is smaller than -1.
Now, if we multiply these two functions: h(x) = f(x) * g(x) = (-x)*(-1/x) = x, the resulting product is h(x) = x.
When we analyze h(x), we find that as x increases, the value of h(x) also increases. For example, when x = 1, h(x) = 1. When x = 2, h(x) = 2, which is greater than 1.
Therefore, we have found an example of two decreasing functions (-x and -1/x), whose product (x) is increasing.