Which of the following is always correct

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If f'(x)>0 ∀ x∈ domain, then f(x) must be one-one

If f'(x)<0 ∀ x∈ domain, then f(x) must be one-one

If |f(x)| be continuous at x=a , then f(x) is also continuous at x=a

If f(x) is continuous at x=a , f(a)=2 and x=a is the point of local minima of f(x) , then [f(x)] , where [.] denotes greatest integer function, is also continuous at x=a

answer is D.

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Detailed Solution

In (A) f'(x)>0 for every x∈Df ⇒ f(x) is one-one ex. f(x)=tanx In (B), f'(x)<0 ∀ x∈Df⇒f is one-oneex. f(x)=cotx In (C), |f(x)| is continuous ⇒ f(x) is continuous ex. f(x) ={1 : x<0−1 : x≥0 is discontinuous at x=0 But |f(x)|=1 ∀x∈R which is continuous ∴ (A), (B), (C) are not correctIn (D), f(a)=2 , x=a is a point of local minimum and f(x) is continuous ∴ f(a+h)>2, f(a−h)>2 ⇒ [f(a+h)]=[f(a−h)]=2 Hence [f(x)] is also continuous at x=a ∴ (D) is correct.

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