Continuity and DifferentiabilityHard
Question
Let [.] denotes the greatest integer function and f(x) = [tan2x], then
Options
A.
f(x) does not exist
f(x) does not existB.f(x) is continuous at x = 0
C.f(x) not differentiable at x = 0
D.f′(0) = 1
Solution
Given, f(x) = [tan2x]
Now, - 45o < x < 45o
⇒ tan (-45o) < tan x < tan (45o)
⇒ - tan 45o < tan (45o)
⇒ - 1 < tan x < 1
⇒ 0 < tan2 x < 1
⇒ [tan2x] = 0
ie, f(x) is zero for all values of x fromx = - 45 to 45o
Thus, f(x) exists when x → 0 and also it is continuous at x = 0
Also, f(x) is differentiable at x = 0 and has a value of zero.
Therefore, (b) is the answer.
Now, - 45o < x < 45o
⇒ tan (-45o) < tan x < tan (45o)
⇒ - tan 45o < tan (45o)
⇒ - 1 < tan x < 1
⇒ 0 < tan2 x < 1
⇒ [tan2x] = 0
ie, f(x) is zero for all values of x fromx = - 45 to 45o
Thus, f(x) exists when x → 0 and also it is continuous at x = 0
Also, f(x) is differentiable at x = 0 and has a value of zero.
Therefore, (b) is the answer.
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