Find the value of k if [-2.3] k – 2k + [8.34] k = [4.23] + [2.23], where [.] represents the smallest integer function.

Concept:

Smallest integer function:

The function f (x) = [x] is called the smallest integer function and it means that the smallest integer is greater than or equal to x i.e [x] ≥ x.

The graph for the smallest integer function is given below,

Example:[3.34] = 4, [0.67] = 1, [4] = 4, [- 8.112] = – 8, [- 0.5] = 0.

In general, if n is an integer and x is any real number between n and (n + 1).

i.e., n < x ≤ n + 1, then [x] = n + 1

CALCULATION:

Given function is f(x) = [-2.3] k – 2k + [8.34] k = [4.23] + [2.23]

As we know that for smallest integer function [-2.3] = -2, [8.34] = 9, [4.23] = 5, [2.23] = 3.

∴ -2k – 2k + 9k = 5 + 3