Let [x] denote the greatest integer function and
f(x)=max {1+x+[x], 2+x, x+2[x]}, 0 ≤ x ≤ 2. Let m be the number of
points in [0,2], where f is not continuous and n be the number of points in
(0,2), where f is not differentiable. Then (m+n)²+2 is equal to :

Solution

Maths sir · 4 steps

Work out [x] on each piece first. Then the max of three lines turns into one simple line.

1. Split by [x]. On [0,2], [x] takes only the values 0, 1 and 2. Put each value into the three expressions and pick the largest.

   Interval	1+x+[x]	2+x	x+2[x]	f(x)
   0 ≤ x < 1, [x]=0	1+x	2+x	x	2+x
   1 ≤ x < 2, [x]=1	2+x	2+x	2+x	2+x
   x = 2, [x]=2	5	4	6	6

2. Check x = 1. On both sides of x=1 the function is the same line 2+x. So f is continuous at x=1, and its slope is 1 on both sides, so it is differentiable there too.
3. Check x = 2. As x → 2⁻, f(x) → 4, but f(2) = 6. So f jumps at x=2, which gives m = 1. Inside (0,2), f(x) = 2+x is a straight line, so n = 0.
4. So. (m+n)² + 2 = 1² + 2 = 3, which is option (a).

The idea to keep. With [x] inside a max or min, fix [x] on each integer interval first. The messy function often collapses to one line, and only the endpoints need checking.

Answer from the NTA final answer key. Solution by PrepWiser.