Problem 34
Question
Show that \(f\) is strictly monotonic on the given interval and therefore has an inverse function on that interval. \(f(x)=\sec x, \quad\left[0, \frac{\pi}{2}\right)\)
Step-by-Step Solution
Verified Answer
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.
1Step 1: 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 \(\sec x \cdot \tan x\).
2Step 2: 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.
3Step 3: Establish Monotonicity and Existence of Inverse Function
As \(f(x)\) is strictly increasing in the interval, it means for any \(x_1
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