Chapter 7: Problem 6
Are the following functions strictly monotonic? (a) \(y=-x^{6}+5 \quad(x>0)\) (b) \(y=4 x^{5}+x^{3}+3 x\) For each strictiy monotonic function, find \(d x / d y\) by the inverse-function rule.
Short Answer
Expert verified
Both functions are strictly monotonic.
\( dx/dy = -\frac{1}{6x^5} \) for (a), and \( dx/dy = \frac{1}{20x^4 + 3x^2 + 3} \) for (b).
Step by step solution
01
Determine Strict Monotonicity of (a)
To determine if the function \( y = -x^6 + 5 \) is strictly monotonic for \( x > 0 \), we need to find its derivative with respect to \( x \). The derivative \( y' \) is given by \[ y' = -6x^5 \]. For \( x > 0 \), \( 6x^5 > 0 \), so \( -6x^5 < 0 \), indicating that the function is strictly decreasing since the derivative is negative for all \( x > 0 \). Thus, the function is strictly monotonic.
02
Determine Strict Monotonicity of (b)
Now consider the function \( y = 4x^5 + x^3 + 3x \). To test for strict monotonicity, we find the derivative \( y' \): \[ y' = 20x^4 + 3x^2 + 3 \]. Each term in this derivative is non-negative for all real \( x \), and except for \( x = 0 \), the derivative will be positive, which means the function is strictly increasing for all \( x \) because \( y' > 0 \) for all \( x eq 0 \). Hence, the function is strictly monotonic.
03
Find Inverse Derivative of (a)
Since the function \( y = -x^6 + 5 \) is strictly monotonic over \( x > 0 \), we can use the inverse-function theorem to find \( \frac{dx}{dy} \). The theorem states \( \frac{dx}{dy} = \frac{1}{dy/dx} \). We already found \( dy/dx = -6x^5 \). So, \( \frac{dx}{dy} = \frac{1}{-6x^5} \).
04
Find Inverse Derivative of (b)
Similarly, for \( y = 4x^5 + x^3 + 3x \), using the inverse-function theorem, \( \frac{dx}{dy} = \frac{1}{dy/dx} \). We found \( dy/dx = 20x^4 + 3x^2 + 3 \). Thus, \( \frac{dx}{dy} = \frac{1}{20x^4 + 3x^2 + 3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inverse Function Theorem
The Inverse Function Theorem is a fundamental concept in calculus that allows us to find derivatives of inverse functions. It provides a formula for the derivative of an inverse function using the derivative of the original function.
Here's how it works in simple terms: if you have a function with a derivative that is non-zero around a point, then its inverse exists and is differentiable at that point. The theorem gives us the formula:
This formula is incredibly useful in finding the rate of change of an inverse function without having to explicitly solve for it.
Imagine we have a function where calculating the output for each input is straightforward, but reversing this process, i.e., finding the input for a given output, is difficult. With the Inverse Function Theorem, we can instead solve for this derivative by using what's simpler to compute: the derivative of the original function.
Here's how it works in simple terms: if you have a function with a derivative that is non-zero around a point, then its inverse exists and is differentiable at that point. The theorem gives us the formula:
- \(\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}\)
This formula is incredibly useful in finding the rate of change of an inverse function without having to explicitly solve for it.
Imagine we have a function where calculating the output for each input is straightforward, but reversing this process, i.e., finding the input for a given output, is difficult. With the Inverse Function Theorem, we can instead solve for this derivative by using what's simpler to compute: the derivative of the original function.
Derivative
The derivative of a function describes the rate at which the function's value changes as its input changes.
In simpler terms, the derivative gives us the slope of the tangent line to the function's graph at any point. For a function \( y = f(x) \), the derivative, denoted by \( f'(x) \) or \( \frac{dy}{dx} \), tells us how y changes concerning x.
For example, consider a function \( y = x^2 \). Its derivative \( \frac{dy}{dx} = 2x \) tells us that, for each unit increase in x, y increases by twice the value of x.
Calculating derivatives is essential in various fields like physics and economics since they allow us to understand how quantities change over time.
In mathematical economics, derivatives help economists model and predict behaviors, like how changes in price might affect demand.
In simpler terms, the derivative gives us the slope of the tangent line to the function's graph at any point. For a function \( y = f(x) \), the derivative, denoted by \( f'(x) \) or \( \frac{dy}{dx} \), tells us how y changes concerning x.
For example, consider a function \( y = x^2 \). Its derivative \( \frac{dy}{dx} = 2x \) tells us that, for each unit increase in x, y increases by twice the value of x.
Calculating derivatives is essential in various fields like physics and economics since they allow us to understand how quantities change over time.
In mathematical economics, derivatives help economists model and predict behaviors, like how changes in price might affect demand.
Strict Monotonicity
Strict monotonicity is a property of functions that describes whether they are consistently increasing or decreasing over their domain.
A function is strictly increasing if, as x values increase, y values also increase continually without exception. Conversely, a function is strictly decreasing if, as x values increase, the y values decrease.
The importance of strict monotonicity lies in its ability to assure us about the function's behavior over the entire domain, making it predictable and ensuring continuity.
If a function is strictly monotonic, it has at most one real root, and an inverse function is guaranteed, as the function is either entirely increasing or decreasing, with no repeated y-values.
This property makes such functions very useful in optimization problems and economic models, where consistent trends in data help in making reliable predictions about the market.
A function is strictly increasing if, as x values increase, y values also increase continually without exception. Conversely, a function is strictly decreasing if, as x values increase, the y values decrease.
The importance of strict monotonicity lies in its ability to assure us about the function's behavior over the entire domain, making it predictable and ensuring continuity.
If a function is strictly monotonic, it has at most one real root, and an inverse function is guaranteed, as the function is either entirely increasing or decreasing, with no repeated y-values.
This property makes such functions very useful in optimization problems and economic models, where consistent trends in data help in making reliable predictions about the market.
Mathematical Economics
Mathematical economics uses mathematical methods, like calculus and algebra, to represent economic theories and analyze problems presented in economics.
The derivative, for instance, is a tool that's heavily used in economics to model marginal concepts, such as marginal cost or marginal revenue.
These models help economists understand how a small change in one variable, like price, can affect other variables, such as the quantity of goods demanded.
In a similar vein, monotonic functions and their inverse relationships are crucial in economics to describe relationships that maintain consistent trends, such as supply and demand curves.
Economists use the concept of monotonicity to ensure that their functions model reality, with price typically being inversely related to demand, and positively related to supply, under normal conditions.
This field allows theoretical predictions to be tested, giving policymakers and businesses tools to evaluate different economic scenarios analytically.
The derivative, for instance, is a tool that's heavily used in economics to model marginal concepts, such as marginal cost or marginal revenue.
These models help economists understand how a small change in one variable, like price, can affect other variables, such as the quantity of goods demanded.
In a similar vein, monotonic functions and their inverse relationships are crucial in economics to describe relationships that maintain consistent trends, such as supply and demand curves.
Economists use the concept of monotonicity to ensure that their functions model reality, with price typically being inversely related to demand, and positively related to supply, under normal conditions.
This field allows theoretical predictions to be tested, giving policymakers and businesses tools to evaluate different economic scenarios analytically.
