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Chapter 5: Problem 33

Show that \(f\) is strictly monotonic on the given interval and therefore has an inverse function on that interval. \(f(x)=\cos x, \quad[0, \pi]\)

Short Answer

Expert verified
The function \(f(x) = \cos x\) is strictly monotonic on the interval [0, 蟺] and it has an inverse function on the same interval.

Step by step solution

01

Define the function and interval

We begin with the function \(f(x) = \cos x\) in the interval \([0, \pi]\).
02

Check monotonicity

To check the monotonicity of the function we have to take derivative of the function. The derivative of \(f(x) = \cos x\) is \(f'(x) = -\sin x\). As we know, \(-\sin x\) over the interval [0, 蟺] is nonpositive, because sine function yields positive results in this interval, and we're negating those results. Hence, \(f'(x)\) is less than or equal to zero for any 'x' in the given interval, which implies that the function is decreasing in this interval. Thus \(f(x)\) is strictly monotonic.
03

Validate strictly monotonic behavior

We can test with few values in the interval of \([0, \pi]\) to validate the strictly monotonically decreasing behavior of the function. For example, at \(x = 0\), \(f(0) = \cos 0 = 1\), and at \(x = \pi\), \(f(\pi) = \cos \pi = -1\). As \(x\) increases from 0 to 蟺, \(f(x)\) decreases from 1 to -1.
04

Inverse

As the function \(f(x)\) is strictly monotonic on the interval [0, 蟺], by the Monotonic Function Theorem, \(f(x)\) has an inverse function on the interval [0, 蟺].

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