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Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = sin x 2 , π 2 , 3π 2

Respuesta :

Answer:

The numbers   [tex]3\pi/2 , 5\pi/2[/tex]   satisfy the conclusion of Rolle's Theorem

Step by step.

1. The function must be continuous.

Trigonometric functions are continuous.

2.  It must be true that [tex]f(a) = f(b) = 0[/tex]

For this case      [tex]sin(\pi) = sin( 3\pi) = 0[/tex]

3. Therefore by Rolle's Theorem, there exist a point   [tex]x[/tex]   such that [tex]f'(x) = 0[/tex].

For this case    [tex]f'(x) = cos(x)[/tex]

And     [tex]cos(x) = 0[/tex]      at      [tex]x = 3\pi/2 , 5\pi/2[/tex]

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