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Use Rolle's Theorem to determine the value of
such that ′ () = 0,
if () = cos
in the interval [- , ].BBB DDD
Step 1: () = cos [Write the function.]
Step 2: (- ) = cos (- ) = 0[Find (- ).]
Step 3: () = cos () = 0[Find ().]
Step 4: Since () is continuous in [- , ] and differentiable in (- , ) also (- ) = () = 0, Rolle's Theorem is applicable.
Step 5: There exists a number in (- , ) such that ′() = 0
Step 6: () = cos ′() = - sin [Find ′().]
Step 7: ′() = - sin [Find ′().]
Step 8: - sin = 0[Equate ′() to zero.]
Step 9: = 0, which lies in the interval (- , )[Solve for .]
Step 10: So, the derivative of the function is zero at = 0 in the interval [- , ].