etermine whether Rolle\'s Theorem can be applied to f on the closed interval [a,
ID: 2857249 • Letter: E
Question
etermine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = 5 cos 2x, [, ] Yes, Rolle's Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f (a) f (b). Correct: Your answer is correct. If Rolle's theorem can be applied, find all values of c in the open interval (a, b) such that f '(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) c =
Explanation / Answer
f(x) = 5 cos 2x , Interval given is [, ]
1) f(x) is continuous in the interval [, ]
2) f(x) is differentiable in the interval (, )
3) f() = f() = 0
hence Rolle's theorem can be applied.
f(x) = 5 cos 2x
==> f '(x) = 5 (-sin(2x)) (2) since d/dx f(g(x)) = f '(g(x)) g '(x) , f(x) = cosx , g(x) = 2x , d/dx xn = n xn-1 , here n = 1
==> f '(x) = -10 sin(2x)
==> f '(c) = -10 sin(2c)
f '(c) = 0 ==> -10 sin(2c) = 0
==> sin(2c) = 0
==> 2c = 0 ,
==> c = 0 , /2
Hence the values of c are 0 , /2
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