Problem 23 - 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. \(f(x)=2-x-x^{3}\)

Step-by-Step Solution

Answer

Yes, the function \(f(x)=2-x-x^{3}\) is strictly monotonic on its entire domain and therefore has an inverse function.

Step 1: Compute the Derivative

The derivative of the function \(f(x)=2-x-x^{3}\) needs to be computed. The derivative is given by \(f'(x)=-1-3x^{2}\).

Step 2: Analyze the Monotonicity

Analyze whether the derivative of the function is always negative or positive, which would therefore confirm whether the function is strictly monotonic or not. \(f'(x)=-1-3x^{2}\) is always negative because the square term (i.e., \(x^2\)) is always positive or zero, so with a negative sign in front it will always produce a non-positive result, and when subtracted with 1, the result will always be negative.

Step 3: Conclude

Since the derivative is always negative, the original function \(f(x)=2-x-x^{3}\) is strictly decreasing in its entire domain. Therefore, it is strictly monotonic and has an inverse function.