Determine if Rolle’s Theorem can be applied to f(x)=sinx on [((pi)/6), (5(pi)/6)] If so, find c.

Question

Determine if Rolle’s Theorem can be applied to f(x)=sinx on [((pi)/6), (5(pi)/6)] If so, find c.

Answer 1

The sine function is differentiable, hence continuous, and continuous outright

y = sin x

sin(5*pi/6) = sin(150) = 1/2

sin(pi/6) = sin(30) = 1/2

so the slope of the secant line is zero

y' = cos x

There exists C in [ pi/6, 5*pi/6] such that f'(C) = 0

cos C = 0

C = pi/2 = 3*pi/6

1*pi/6 < 3*pi/6 < 5*pi/6