if f(x)=3sin2x is continous over interval [0,π ] and differential over interval (0,π ) then by rolles theorem the value of c is_____
a)π
b)2
c)π/2
d)π/8
Answers
Answer:
c
Step-by-step explanation:
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Answer:
c = π/4
Step-by-step explanation:
It is given that f(x) = 3sin 2x is continous on [0,π]
We also know that it is differentiable over (0,π )
We know that for Rolle's Theorem to hold true, f(a) = f(b)
Here, f(0) = 3 sin(2*0) = 3 * sin 0 = 3*0 = 0
and f(π) = 3 sin (2* π) = 3 * 0 = 0
As f(a) = f(b) , Rolle's Theorem holds true for the given function.
Thus, f'(c) = 0
Therefore, f(c) = 3sin 2c
f'(c) = 3*2*cos2c
=> 3*2*cos2c = 0
=> cos 2c = 0
=> 2c =
Substituting the given options in c
a) cos 2*π = 1 ≠ 0
b) cos 2*(π/4) = cos (π/2) = 0 satisfied.
c) cos 2*(π/2) = cos π = -1 ≠ 0
d) cos 2 *(π/8) = cos (π/4) = 1/√2 ≠ 0
Therefore, as only c = π/4 can satisfy the given equation, option (b) has to be c = π/4