Question

Suppose that \(f^{\prime}(x)>0\) and \(g^{\prime}(x)>0\) for all \(x .\) What simple additional conditions (if any) are needed to guarantee that: (a) \(f(x)+g(x)\) is increasing for all \(x\); (b) \(f(x)=g(x)\) is increasing for all \(x\); (c) \(f(g(x))\) is increasing for all \(x\) ?

Short answer

(a) No additional conditions needed; (b) Needs more context; (c) Ensure \(f'(g(x)) > 0\).

Step-by-step solution

Analyze Condition for (a)

For part (a), we need to determine the condition under which the sum of two functions \(f(x)\) and \(g(x)\) is increasing. From the problem, we know that \(f'(x) > 0\) and \(g'(x) > 0\) for all \(x\). To ensure that their sum \(f(x) + g(x)\) is increasing, we need the derivative of the sum to be greater than zero. The derivative of the sum is \[(f(x) + g(x))' = f'(x) + g'(x)\]. Since both \(f'(x)\) and \(g'(x)\) are greater than zero, the derivative of their sum is \(f'(x) + g'(x) > 0\) for all \(x\). Hence, no additional conditions are needed to guarantee that \(f(x) + g(x)\) is increasing.

Analyze Condition for (b)

For part (b), we need to ensure that the function \(f(x) = g(x)\) is increasing. This means we want the equality point of \(f(x)\) and \(g(x)\) (if it exists) to shift towards higher \(x\). Since \(f'(x) > 0\) and \(g'(x) > 0\) for all \(x\), these functions are individually increasing. However, to consider the equation \(f(x) = g(x)\) as increasing, we would need to check the behavior of \(h(x) = f(x) - g(x)\). For \(f(x) = g(x)\) to be increasing, \(h'(x) = f'(x) - g'(x) > 0\) must hold or be balanced in such a way to make it increasing, but as written this may not apply generally without the specific context of what it means for the equality function to increase.

Analyze Condition for (c)

For part (c), we need to determine the conditions for the composite function \(f(g(x))\) to be increasing. We apply the chain rule to find the derivative of the composite function: \((f(g(x)))' = f'(g(x)) \cdot g'(x)\). We know from the problem statement that \(g'(x) > 0\). For \(f(g(x))\) to be increasing, we also require \(f'(g(x)) > 0\) for all \(x\). Therefore, we need \(f(x)\) to be increasing for all values in the range of \(g(x)\). Assuming \(f'(y) > 0\) for all \(y\), the function \(f(g(x))\) is guaranteed to be increasing.

Key concepts

Differentiation

Differentiation is a key concept in calculus that involves calculating the derivative of a function. The derivative provides crucial information about the function's rate of change. In simpler terms, it tells us how a function's value changes with a small change in its input variable. Understanding differentiation allows us to determine where a function is increasing or decreasing.

For instance, if a function's derivative, denoted as \(f'(x)\), is greater than zero, the function is increasing at that point. Conversely, if \(f'(x)\) is less than zero, the function is decreasing. In the context of increasing functions, we often seek conditions where the derivative is consistently positive to ensure the function continuously rises as \(x\) increases.

Differentiation is a powerful tool for analyzing and predicting behavior of functions, especially when combined with other rules and methods, such as the chain rule, to examine more complex scenarios like composite functions.

Chain Rule

The chain rule is an essential technique in calculus for differentiating composite functions. A composite function is a function that is formed when one function is applied to the result of another function. The chain rule helps us find the derivative of such functions.

The formula is straightforward: If you have a composite function \(f(g(x))\), the derivative \((f(g(x)))'\) is found by multiplying the derivative of \(f\) evaluated at \(g(x)\) by the derivative of \(g\) with respect to \(x\). Mathematically, this is expressed as \((f(g(x)))' = f'(g(x)) \cdot g'(x)\).

This rule is particularly useful when you have two or more nested functions and need to determine their rate of change. It allows us to break down complex functions into manageable parts, analyze each part's change rate, and understand how these parts contribute to the overall behavior of the composite function. As a result, using the chain rule combined with information about individual functions' derivatives can help determine if a composite function is increasing or decreasing.

Composite Functions

Composite functions, as mentioned, are formed when one function is composed with another. This means inserting the output of one function, \(g(x)\), into another function, \(f\), to create \(f(g(x))\).

Understanding how to deal with composite functions is important in calculus as they frequently appear in real-world problems. To analyze these functions, we often use the chain rule to find their derivatives. By knowing the derivatives of \(f\) and \(g\), we can determine if the composite function \(f(g(x))\) is increasing across its domain.

To ensure a composite function is increasing, conditions on the derivatives of \(f\) and \(g\) must be met. Specifically, if both \(f'(y) > 0\) for all \(y\) in the range of \(g(x)\) and \(g'(x) > 0\) for all \(x\), then \(f(g(x))\) is increasing. This involves checking that the inner function \(g(x)\) is feeding into \(f\) in a manner that respects \(f\)'s increasing nature. Thus, understanding composite functions and their derivatives gives us the power to analyze dynamic systems and changes.