Problem 33 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

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

Step 1: Define the function and interval

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

Step 2: 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, π] 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.

Step 3: 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 π, \(f(x)\) decreases from 1 to -1.

Step 4: Inverse

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