How do you determine if rolles theorem can be applied to # f(x) = sin 2x# on the interval [0, (pi/2)] and if so how do you find all the values of c in the interval for which f'(c)=0?

Question

How do you determine if rolles theorem can be applied to # f(x) = sin 2x# on the interval  [0, (pi/2)] and if so how do you find all the values of c in the interval for which f'(c)=0?

Ezra Adams · Accepted Answer

you do this by determining whether the hypothses for Rolle's Theorem (the "If" parts) are true for  # f(x) = sin 2x# on the interval  [0, (pi/2)] Hypothesis 1: we want:  #f# is continuous on the closed interval #[a,b]# So, is #sin2x# continuous on #[0, pi/2]#?  Why should the reader agree with your answer? Hypothesis 2:  we want:  #f# is differentiable on the open interval #(a,b)# So, is #sin2x# differentiable on #(0, pi/2)#?  Why should the reader agree with your answer? Hypothesis 3:  we want #f(a) = f(b)# So, is #sin(2(0)) = sin (2( pi/2))#?  Why should the reader agree with you? To find all values of #c# in the interval for which #f'(c) = 0#, find #f'(x)#, set it equal to #0# and solve the equation, ignoring solutions outside the interval.

Christopher Agresta · Answer

undefined To determine if Rolle's Theorem can be applied to $ f(x) = \sin(2x) $ on the interval $ [0, \frac{\pi}{2}] $, check if the following conditions are met:

1. The function $ f(x) $ is continuous on the closed interval $ [0, \frac{\pi}{2}] $.
2. The function $ f(x) $ is differentiable on the open interval $ (0, \frac{\pi}{2}) $.
3. $ f(0) = f\left(\frac{\pi}{2}\right) $.

If all these conditions are satisfied, then Rolle's Theorem can be applied. To find all the values of $ c $ in the interval $ [0, \frac{\pi}{2}] $ for which $ f'(c) = 0 $, differentiate $ f(x) $ with respect to $ x $, then set the derivative equal to zero and solve for $ c $.

For $ f(x) = \sin(2x) $:
$$ f'(x) = 2\cos(2x) $$

Setting $ f'(x) = 0 $ and solving for $ x $, we get:
$$ 2\cos(2c) = 0 $$
$$ \cos(2c) = 0 $$

This equation has solutions in the interval $ [0, \frac{\pi}{2}] $ when $ 2c = \frac{\pi}{2} $ or $ 2c = \frac{3\pi}{2} $.

So, the values of $ c $ where $ f'(c) = 0 $ in the interval $ [0, \frac{\pi}{2}] $ are:
$$ c_1 = \frac{\pi}{4} $$
$$ c_2 = \frac{3\pi}{4} $$