Problem 26 - Calculus of a Single Variable

Question

Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function. \(f(x)=x^{5}+2 x^{3}\)

Step-by-Step Solution

Answer

The function \(f(x)=x^{5}+2 x^{3}\) is not strictly monotonic on its entire domain hence it does not have an inverse over its whole domain.

Step 1: Compute the derivative

The derivative of \(f(x)=x^{5}+2 x^{3}\) is given by \(f'(x)=5x^{4}+6x^{2}\).

Step 2: Check the sign of the derivative

The derivative \(f'(x)=5x^{4}+6x^{2}\) is a sum of two squares, hence it is always non-negative. However, there exist an x-value, x=0, where the derivative equals zero. Therefore, the function is not strictly increasing or decreasing across its entire domain, thus it doesn't have an inverse over its whole domain.

Step 3: Determine the monotonicity of the function

Having established that \(f'(x)=5x^{4}+6x^{2}\) is not consistently positive or negative across all x, this indicates that the function \(f(x)=x^{5}+2 x^{3}\) is not strictly monotonic.

Key Concepts

Strictly Monotonic Functions

A strictly monotonic function is either entirely increasing or decreasing over its domain. It never stays constant, meaning its output consistently goes up or down as the input changes. This property is crucial for the function to have an inverse across its entire domain. If a function changes direction even once, it breaks the criteria of strict monotonicity.
In simpler terms, a strictly increasing function means that for any two numbers, say \(x_1\) and \(x_2\), whenever \(x_1 < x_2\), then \(f(x_1) < f(x_2)\). On the other hand, a strictly decreasing function has the condition that \(f(x_1) > f(x_2)\) if \(x_1 < x_2\).
This consistency in the function’s behavior ensures that every output is paired with a unique input. This unique pairing allows us to "reverse" the function, and hence, find an inverse.

Derivative Test

The derivative test is a powerful method to determine if a function is strictly monotonic. To put it into practice, you'll take the derivative of the function, \(f'(x)\), and examine its sign. The sign of the derivative tells us how a function behaves over its domain.

• If \(f'(x) > 0\) for all \(x\), then the function is strictly increasing.
• If \(f'(x) < 0\) for all \(x\), then the function is strictly decreasing.

If the derivative is zero or does not maintain a consistent sign, it indicates that the function is not strictly monotonic across its entire domain.
In the exercise, we found that the derivative of \(f(x) = x^5 + 2x^3\) was \(f'(x) = 5x^4 + 6x^2\). Since this is always non-negative and includes points where it equals zero, the function is not strictly monotonic. Its behavior doesn't fully satisfy the requirements to have an inverse function available on its entire domain.

Function Monotonicity

Function monotonicity refers to the nature of a function's rise or fall over an interval. While strict monotonicity requires a function to consistently increase or decrease without plateauing, monotonicity, in general, allows a function to be non-decreasing or non-increasing.
Non-decreasing implies the function doesn't fall, but it can remain constant for some intervals. Similarly, non-increasing means it doesn't rise, yet can stay unchanged in certain regions. Thus, monotonic functions can include intervals where the function's value doesn't change.

• A function is monotonic if it is either entirely non-decreasing or non-increasing over its domain.
• This characteristic helps predict the behavior of function values without strictly demanding constant change.

The function \(f(x) = x^5 + 2x^3\), as previously analyzed, is neither strictly increasing nor strictly decreasing due to the derivative's behavior. It's crucial to understand that without strict monotonicity, a function might not possess an inverse over its entire domain.