Ans. 0030

Sol. f(x) = [x] + [2x] + + [3x] +[4x] + [5x] [kx] changes its value of every integral multiple of 1/k

[x] will change at every integral multiple of 1

[2x] will change at every integral multiple of ½

[3x] will change at every integral multiple of 1/3

[4x] will change at every integral multiple of ¼

[5x] will change at every integral multiple of 1/5

and will change at every integral multiple of 3/2

They would change all together at every multiple of LCM of {1, 1/2, 1/3, 1/4, 1/5, 3/2} = 3

No. of total points at which f(x) will changes its value in the interval [0, 3] will depend on the total number of different terms in the following cases –

[x] = 0, 1, 2

[2x] = 0, 1/2, 2/2, 3/2, 4/2, 5/2

[3x] = 0, 1/3, 2/3, 3/3, 4/3, 5/3, 6/3, 7/3, 8/3

[4x] = 0, 1/4, 2/4, 3/4, 4/4, 5/4, 6/4, 7/4, 8/4, 9/4, 10/4

[5x] = 0, 1/5, 2/5, 3/5, 4/5, 5/5, 6/5, 7/5, 8/5, 9/5, 10/5, 11/5, 12/5, 13/5, 14/5

= 0, 3/2, 6/2

∴ f(x) will change its values in the intervals

0 ≤ x < 1/5, 1/5 ≤ x < 1/4, 1/4 ≤ x < 1/3,........... ≤ x < 3

Total no. of different terms in above equation = 30 so no. of

terms in the range of

f(x) for 0 ≤ x < 3 is 30 ⇒ 0030