Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The function \(f\) is increasing and concave up on \((-\infty, \infty) \cdot f^{\prime}(x)\) is never zero. Denote by \(g(x)\) the inverse of \(f\). (a) What is the sign of \(g^{\prime}\) ? (b) What is the sign of \(g^{\prime \prime}\) ? (c) If \(f(3)=5\) and \(f^{\prime}(3)=10\), what is \(g^{\prime}(5) ?\)

Step-by-Step Solution

Answer

Part (a), \(g^{\prime}(x) > 0\). Part (b), the sign is undetermined. Part (c), \(g^{\prime}(5) = 0.1\).

Step 1: Sign of the derivative of the Inverse function

To find the sign of \(g^{\prime}(x)\), the derivative of the inverse function, we need to use the derivative of the inverse function theorem. According to which, if \(f\) is differentiable and increasing on an interval \(I\) and \(f^{\prime}(x) != 0\) for all \(x\) in \(I\), then the derivative of the inverse function \(f^{-1}\) at \(x\) is \(1/(f^{\prime}(f^{-1}(x))\). Since \(f\) is increasing, \(f^{\prime}(x) > 0\), hence \(g^{\prime}(x) = 1/(f^{\prime}(f^{-1}(x))) > 0\).

Step 2: Sign of the second derivative of the Inverse function

To find the sign of the \(g^{\prime \prime}(x)\), the second derivative of the inverse function, we need to use second derivative of inverse function formula. But since we do not have the second derivative values of function \(f\), we cannot deduce the sign of the the second derivative of \(g\). So, the answer is: we cannot determine it.

Step 3: Derivative of the inverse function for a given value

To find the value of \(g^{\prime}(5)\), we use the fact that \(f(3) = 5\) and \(f^{\prime}(3) = 10\). By setting \(y = f(x)\), where \(x = f^{-1}(y)\), if \(y = 5\), then \(x = 3\). According to the theorem of derivative of inverse function, it follows that \(g^{\prime}(5) = 1/f^{\prime}(3) = 1/10 = 0.1\).

Key Concepts

Derivative of Inverse Functions

Understanding the derivative of inverse functions can be very helpful, especially when dealing with problems that involve functions and their inverses.
When we say that a function has an inverse, it means that for every output of the function, there is one specific input that produced it.
To find the derivative of an inverse function, we use the theorem: if a function \(f\) is differentiable and increasing, and its derivative \(f'(x) eq 0\), then the derivative of its inverse \(g(x) = f^{-1}(x)\) is given by:

• \(g'(x) = \frac{1}{f'(f^{-1}(x))}\)

This theorem simplifies the process of finding the derivative of the inverse function by not requiring us to explicitly determine the inverse function first.
Knowing this formula is crucial because it tells us about the behavior of the inverse function based on the behavior of the original function.
If \(f(x)\) is increasing and \(f'(x) > 0\), then the inverse function \(g(x)\) must also be increasing because \(g'(x) > 0\). This concept is powerful as it allows us to infer properties about an inverse function without directly computing it.

Concave Up Functions

A function being concave up implies that its second derivative is positive.
This is an important characteristic to understand as it describes how the shape of the function behaves.
If a function \(f(x)\) is concave up on an interval, any tangent line to the function will lie below the curve of the function. This indicates that the slope of \(f(x)\) is increasing across that interval.
Let's consider why this is particularly relevant for inverse functions. When you take the inverse of a function that is concave up and increasing, the behavior can affect the second derivative of the inverse. However, calculating the exact behavior of \(g''(x)\) for an inverse function is complex without further information about additional higher-order derivatives of the original function.

• This characteristic tells us that the original function not only rises but does so increasingly faster, indicating acceleration in growth.
• Understanding if a function is concave up helps predict how the function increases, providing deeper insight into the behavior of its inverse.

In practical terms, knowing if the original function is concave up aids in mathematical modeling and anticipating how changes affect real-world phenomena modeled by such equations.

Increasing Functions

When we describe a function as increasing, it means that as we move from left to right on the graph, the function values get larger.
This kind of behavior is fundamental in calculus, especially for understanding inverse functions and their derivatives.
For the function \(f(x)\) to be increasing, its derivative \(f'(x)\) must be positive for all \(x\) in its domain.

• This information tells us that \(f(x)\) is continuously moving upwards, which directly influences the derivative of its inverse, making it positive.
• When \(f(x)\) is strictly increasing and differentiable, its derivative does not equal zero, which guarantees the existence of an inverse function.

Knowing that a function is increasing also has practical implications.
For instance, in questions related to rates of change and growth, knowing that the base function increases continuously allows us to conclude that these attributes mirror in its inverse as well, assuming it exists.
Conceptually, this contributes to understanding not just mathematical functions, but any process that can be modeled with similar equations, such as economics, physics, and biology, where increase and rate changes are commonly analyzed.