Please find the error(s) in the proof and provide an explanation in a short paragraph why the reasoning is incorrect.

Theorem used in the incorrect solution:

2.4.4

Find the error(s) in the following "Proof", and, as usual, provide an explanation. Statement The function | cos (@) | is constant on the interval [0, ]. Proof Let F(x) = | cos(x)|. Note that +cos(x) = - sin(x) and sin(@) > 0 if 0 0 it follows that dF d cos (I) 1 = - sin(x) ], it follows that the maximum and minimum values of F occur at the endpoints of the interval [0, ]. But F(0) = | cos(0)| = |1| = 1 and F'(7) = | cos()| = | - 1 = 1. In other words, the maximum and minimum values of F on the interval [0, 7] are 1. It follows that F has constant value 1 on this interval.DEFINITION 4.3.3 CRITICAL POINT The interior points c of the domain of f for which f'(c) =0 or f'(c) does not exist are called the critical points for f.THEOREM 4.3.2 Suppose that c is an interior point of the domain of f. If f has a local maximum or local minimum at c, then f' (c) = 0 or f (c) does not exist.THEOREM 2.4.4 If g is continuous at c and f is continuous at g(c), then the composition fog is continuous at c.THEOREM 2.6.2 THE EXTREME-VALUE THEOREM If f is continuous on a bounded closed interval [a, b], then on that interval f takes on both a maximum value M and a minimum value m