Suppose that f(x) and g(x) are differentiable and negative everywhere. If f^'(x)>0 and g^'(x)>0

step 1 Okay, so we're assuming that for any numbers W and X, we have F of W minus F of X, is less than

Suppose that f(x) and g(x) are differentiable and negative...

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Key Concepts

Product Rule

The product rule is a fundamental technique in differential calculus used to differentiate products of two functions. It states that the derivative of a product, h(x) = f(x)g(x), equals f'(x)g(x) + f(x)g'(x). This concept is key when analyzing how the rate of change of the product function relates to the derivatives and sign properties of its factors.

Monotonicity and Derivative Sign

Monotonicity refers to the behavior of functions—being either increasing or decreasing—over intervals. A function is decreasing if its derivative is negative throughout its domain. In this context, understanding that the signs of f(x), g(x), and their derivatives determine the sign of the product function's derivative is crucial for concluding that h(x) is decreasing.

Sign Analysis in Differentiation

Sign analysis involves examining the positive or negative nature of functions and their derivatives. Here, noting that both f(x) and g(x) are negative everywhere, while their derivatives f'(x) and g'(x) are positive, is essential. When these signs are substituted into the product rule, each term becomes negative, confirming that h'(x) is negative, and thus h(x) is a decreasing function.

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