Problem 28 - 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. \(f(x)=\cos \frac{3 x}{2}\)

Step-by-Step Solution

Answer

No, the function \(f(x) = \cos(\frac{3x}{2})\) is not strictly monotonic on its entire domain, therefore, it does not have an inverse function.

Step 1: Find the Derivative of the Function

To begin, find the derivative of \(f(x) = \cos(\frac{3x}{2})\). Apply the Chain Rule, which proposes that the derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function. The derivative of \(\cos(x)\) is \(-\sin(x)\), and the derivative of \(\frac{3x}{2}\) is \(\frac{3}{2}\). So, the derivative of the function is \(f'(x) = -\sin\left(\frac{3x}{2}\right) * \frac{3}{2} = -\frac{3}{2}\sin\left(\frac{3x}{2}\right)\).

Step 2: Analyze the Derivative of the Function

\(-\frac{3}{2}\sin\left(\frac{3x}{2}\right)\) is not always positive or always negative since the sine function varies between -1 and 1, which means the original function is not strictly increasing or decreasing on its entire domain.

Step 3: Conclude if the function has an inverse

Since the function \(f(x) = \cos(\frac{3x}{2})\), is not strictly increasing or decreasing, this indicates the function is not strictly monotonic. Due to this, it does not meet the criteria for having an inverse function on its entire domain.