MonotonicityHard
Question
If f(0) = f(1) = f(2) = 0 & function f(x) is twice differentiable in (0, 2) and continuous in [0, 2]. Then which of the following is/are definitely true -
Options
A.f”(c) = 0 ; ∀ c∈ (0, 2)
B.f′(c) = 0; for at least two c ∈ (0, 2)
C.f′(c) = 0 ; for exactly one c ∈ (0, 2)
D.f′(c) = 0 ; for at least one c ∈ (0, 2)
Solution
∵ f (0) = f (1)& ′f′ is continuous in [0, 1] derivable in (0, 1)
∴ f′(c ) = 0 for at least one c ∈ (0, 1)
similarly, ∵ f(1) = f(2)
∴ f′(c) = 0 for at least one c2 ∈ (1,2)
⇒ f′(c) = f′(c)
⇒ f″(c) = 0 for at least one e ∈ (c1, c2)
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