If [t] denotes the greatest integer ≤ t, then the number of points, at which the function f(x)=4|2 x+3|+9[x+1/2]-12[x+20] is not differentiable in the open interval (-20,20), is __________.

Solution

Maths sir · 3 steps

Each floor jump breaks the function, so count both sets of jumps, then check the modulus corner.

1. Tidy the function. Since 20 is an integer, [x+20] = [x] + 20. So f(x) = 4|2x+3| + 9[x+1/2] - 12[x] - 240.
2. Jumps of each part. [x] jumps only at integers. [x+1/2] jumps only where x+1/2 is an integer, that is, at half-integers like -3/2, -1/2, 1/2. These two sets never overlap, so one jump can never cancel the other, and at every such point f is discontinuous, hence not differentiable.

   Source	Points in (-20, 20)	Count
   -12[x]	x = -19, -18, …, 19	39
   9[x+1/2]	x = -19.5, -18.5, …, 19.5	40
   4|2x+3|	x = -3/2, already a half-integer	0 new

3. So. The total is 39 + 40 = 79, so f is not differentiable at 79 points.

The common slip: 80. Counting the modulus corner x = -3/2 as an extra point gives 80. It is already one of the 40 half-integers.

The idea to keep. A greatest integer term with a nonzero coefficient breaks the function at every jump. Count those jumps first, then add a modulus corner only if it falls at a new point.

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