Where [.] represents greatest integer function and {.} represents fractional part of a real number then

• (a) λ = 1 ∀ x ∈ R
• (b) λ = 0 ∀ x ∈ R
• (c) λ = 1 ∀ {x} ≥ ½
• (d) λ = 0 ∀ {x} < ½

The question involves understanding the behavior of the greatest integer function and the fractional part of a real number. Let's break down the options regarding the value of λ based on the conditions provided.

Definitions

The greatest integer function, denoted as [x], gives the largest integer less than or equal to x. The fractional part, denoted as {x}, is defined as {x} = x - [x], which represents the non-integer part of x.

Analyzing the Options

• (a) λ = 1 ∀ x ∈ R: This suggests that λ is always 1 for any real number x. This is not true since λ depends on the fractional part.
• (b) λ = 0 ∀ x ∈ R: This implies that λ is always 0 for any real number x. This is also incorrect as λ can vary based on the value of {x}.
• (c) λ = 1 ∀ {x} ≥ ½: This means that if the fractional part of x is greater than or equal to 0.5, then λ equals 1. This is a valid condition since {x} can indeed take values in this range.
• (d) λ = 0 ∀ {x}: This suggests that λ is always 0 for any fractional part, which is not accurate as {x} can take values from 0 to just below 1.

Conclusion

Among the options, (c) λ = 1 ∀ {x} ≥ ½ is the most accurate statement, as it correctly identifies a condition under which λ takes a specific value based on the fractional part of x.