Let f : R → R be a function defined as f(x) = { 3( 1 - |x|/2 ), if, |x| ≤ 2; 0, if, |x| > 2
Let g : R → R be given by g(x) = f(x + 2) - f(x - 2). If n and m denote the number of points in R where g is not continuous and not differentiable, respectively, then n + m is equal to ______________.

Solution

Maths sir · 4 steps

Draw g as two shifted triangles and check the slope on each side of every corner, especially at 0.

1. Shape of f. f is a triangle: it rises from 0 at x=-2 to 3 at x=0 and falls back to 0 at x=2. It is continuous everywhere, so f(x+2), f(x-2) and their difference g are all continuous. That gives n = 0.
2. Write g piecewise. f(x+2) is the triangle moved to sit on [-4, 0], and -f(x-2) is an upside-down triangle on [0, 4]. Each piece of g is a straight line.

   Interval	g(x)	Slope
   x < -4	0	0
   -4 ≤ x ≤ -2	3/2(x+4)	3/2
   -2 ≤ x ≤ 0	-3/2x	-3/2
   0 ≤ x ≤ 2	-3/2x	-3/2
   2 ≤ x ≤ 4	3/2(x-4)	3/2
   x > 4	0	0

3. Compare slopes. The slope changes at x = -4, -2, 2, 4, so g has a corner there. At x = 0 the slope is -3/2 on both sides: the two corners cancel and g is differentiable there. So m = 4.
4. So. n + m = 0 + 4 = 4.

The common slip: 5. Counting x = 0 as a corner because both shifted triangles have a kink there gives 5. The left and right slopes of g at 0 are both -3/2.

The idea to keep. When you add or subtract functions that have corners at the same point, check the slopes on both sides of that point. Two corners can cancel and leave the sum smooth.

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