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Chapter 5: Problem 34

Show that \(f\) is strictly monotonic on the given interval and therefore has an inverse function on that interval. \(f(x)=\sec x, \quad\left[0, \frac{\pi}{2}\right)\)

Short Answer

Expert verified
The function \(f(x)=\sec x\) is strictly monotonic on the interval \([0, \frac{\pi}{2})\), and therefore has an inverse function on that interval.

Step by step solution

01

Calculate Derivative

Firstly, we need to find the derivative of the function \(f(x)\). This is done using the chain rule. The derivative of sec(x) is \(\sec x \cdot \tan x\).
02

Analyze the Derivative

In the given interval \([0,\frac{\pi}{2})\), both \(\sec x\) and \(\tan x\) are positive. Since two positive numbers multiplied resultant in another positive number, we can conclude that the derivative of \(f(x)\) is positive in the given interval. This signifies the function is strictly increasing.
03

Establish Monotonicity and Existence of Inverse Function

As \(f(x)\) is strictly increasing in the interval, it means for any \(x_1<x_2\) in \([0, \frac{\pi}{2})\), \(f(x_1)<f(x_2)\). Hence, \(f\) is strictly monotonic on \([0, \frac{\pi}{2})\). Therefore, according to the monotonic function theorem, it has an inverse function on that interval.

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