Question

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 2

• A. 2
• B. 3
• C. 6
• D. 11

Answer 1

The Correct Option is B

Let's analyze the function \(f(x) = \max\{1+x+[x], 2+x, x+2[x]\}\) over the interval \(0 \leq x \leq 2\).

The greatest integer function \([x]\) denotes the largest integer less than or equal to \(x\). Thus, \([x]\) takes integer values.

Consider the sub-intervals on \([0, 2]\) based on the value of \([x]\): 

• If \(0 \leq x < 1\), then \([x] = 0\).
• If \(1 \leq x < 2\), then \([x] = 1\).
• If \(x = 2\), then \([x] = 2\).

We will evaluate \(f(x)\) in each sub-interval:

1. For \(0 \leq x < 1\):
   • \(1 + x + [x] = 1 + x + 0 = 1 + x\)
   • \(2 + x\)
   • \(x + 2[x] = x + 0 = x\)
2. For \(1 \leq x < 2\):
   • \(1 + x + [x] = 1 + x + 1 = 2 + x\)
   • \(2 + x\)
   • \(x + 2[x] = x + 2 = x + 2\)
3. For \(x = 2\):
   • \(1 + x + [x] = 1 + 2 + 2 = 5\)
   • \(2 + x = 4\)
   • \(x + 2[x] = 2 + 4 = 6\)

Now, identify the points where \(f(x)\) is not continuous or differentiable:

• Continuity check at \(x = 1\): As \(f(x)\) transitions between forms, it maintains continuity because \(2 + x = x + 2\) at \(x = 1\).
• Differentiability check:
  • 0 ≤ x < 1: \(f(x) = 2 + x\), derivative is 1.
  • 1 ≤ x < 2: \(f(x) = x + 2\), derivative is 1.

Therefore, the function \(f(x)\) has:

• \(m = 1\) non-continuous point at \(x = 2\).
• \(n = 1\) non-differentiable point at \(x = 1\).

Thus, \((m+n)^2 + 2 = (1+1)^2 + 2 = 4 + 2 = 6\).

Hence, the correct answer is 6 and not 3. However, as it seems a typo in options or answer key should have 6 as the correct answer.