Question

If \(\mathrm{f}^{\prime}(\mathrm{x})>0\) and \(\mathrm{g}^{\prime}(\mathrm{x})<0 \mathrm{x} \in \mathrm{R}\) then (a) \(\mathrm{f}(\mathrm{g}(\mathrm{x}))>\mathrm{f}(\mathrm{g}(\mathrm{x}+1))\) (b) \(f(g(x))g(f(x+1))\) (d) \(\mathrm{g}(\mathrm{f}(\mathrm{x}))>\mathrm{g}(\mathrm{f}(\mathrm{x}-1))\)

Short answer

The short answer based on the given solution is: Statement (a) and statement (c) are true, while statement (b) and statement (d) are false.

Step-by-step solution

Statement (a): \(f(g(x))>f(g(x+1))\)

Take two real numbers, \(x\) and \(x+1\). Since g is a decreasing function (\(g′(x)<0\)), we have \(g(x) > g(x+1)\). Now, since f is an increasing function (\(f'(x)>0\)), if the input increases, the output also increases. Thus, we have \(f(g(x)) > f(g(x+1))\). So, statement (a) is true.

Statement (b): \(f(g(x))

This statement is the opposite of statement (a). Since we have proven statement (a) to be true, statement (b) must be false.

Statement (c): \(g(f(x))>g(f(x+1))\)

Take two real numbers, \(x\) and \(x+1\). Since f is an increasing function (\(f'(x)>0\)), we have \(f(x) < f(x+1)\). Now, since g is a decreasing function (\(g'(x)<0\)), if the input increases, the output decreases. Thus, we have \(g(f(x)) > g(f(x+1))\). So, statement (c) is true.

Statement (d): \(g(f(x))>g(f(x-1))\)

This statement is the opposite of statement (c) with \(x-1\) instead of \(x+1\). Since we have proven statement (c) to be true, statement (d) must be false.

Key concepts

Increasing Function

An increasing function is a fundamental concept in calculus and algebra. A function \( f(x) \) is said to be increasing if, as the input \( x \) increases, the output \( f(x) \) also increases. This implies that the function moves upwards as you move from left to right on a graph.

To determine whether a function is increasing, we can check its derivative. If the derivative, \( f'(x) \), is greater than zero at every point in its domain, then \( f(x) \) is an increasing function. For example:

• The function \( f(x) = 2x \) is increasing because its derivative \( f'(x) = 2 \) is always greater than zero.
• Functions like \( f(x) = x^2 \) are not always increasing. For \( x < 0 \), the function decreases, but for \( x > 0 \), it increases.

This characteristic of increasing functions is used in calculus to understand the behavior of composite functions, such as those involved in the composition of \( f(g(x)) \).

Decreasing Function

Understanding a decreasing function is similar, but opposite, to an increasing function. A function \( g(x) \) is decreasing if, as the input \( x \) increases, the output \( g(x) \) decreases. On a graph, this function moves downward as you move from left to right.

The derivative of a decreasing function is less than zero. Let's break it down:

• If \( g'(x) < 0 \) for all \( x \) in the domain, then \( g(x) \) is decreasing.
• An example of such a function is \( g(x) = -x \), where \( g'(x) = -1 \). Since \(-1 < 0\), the function is decreasing across its entire domain.

The property of being decreasing is vital in analyzing function compositions like \( g(f(x)) \), where the decrease in one function can significantly influence the outcome of composite functions.

Function Composition

Function composition combines two or more functions to form a new function. You can think of it like nesting one function inside another. For the functions \( f \) and \( g \), the composition \( f(g(x)) \) means you first apply \( g \) to \( x \), and then apply \( f \) to the result of \( g(x) \).

Let's consider an example:

• If \( f(x) = 2x \) (which is increasing) and \( g(x) = -x \) (which is decreasing), then \( f(g(x)) = 2(-x) = -2x \).
• This new function \( -2x \) itself shows the compounded effect of both functions.

In this context, it is important to analyze how monotonicity, whether increasing or decreasing, affects the composition, as seen in problems like selecting the correct statements in the original exercise.

Derivative Sign Analysis

Derivative sign analysis is a technique that helps in understanding the behavior of functions, especially in identifying whether a function is increasing or decreasing. By looking at the sign of the derivative \( f'(x) \), one can determine:

• Positive derivative: The function is increasing at that interval.
• Negative derivative: The function is decreasing at that interval.

Consider the original exercise: the derivative conditions \( f'(x) > 0 \) and \( g'(x) < 0 \) played a crucial role in determining the truth of statements about the composite functions \( f(g(x)) \) and \( g(f(x)) \).

Through derivative sign analysis, you can predict the behavior of function compositions without graphing them. This method simplifies understanding complex problems, making calculus concepts significantly easier to grasp for students.