give an example of two decreasing functions whose product is increasing
To find an example of two decreasing functions whose product is increasing, let's start by understanding the condition for a product of two functions to be increasing.
The product of two functions, denoted as f(x) and g(x), is increasing if the rate of change of their product is positive. Mathematically, this can be expressed as:
(f(x)g(x))' > 0
Differentiating both sides of this inequality, we get:
f'(x)g(x) + f(x)g'(x) > 0
Now, let's consider an example:
Let's take f(x) = -x and g(x) = -1/x. Both of these functions are decreasing.
Now, let's find the derivative of their product and simplify the expression:
(f(x)g(x))' = (-x) * (-1/x) + (-1/x) * (-x) = 1 + 1 = 2
Since the derivative of their product is a positive constant (2), it implies that their product is increasing.
Hence, an example of two decreasing functions whose product is increasing is f(x) = -x and g(x) = -1/x.