Question

Let \( f(x) \) be defined as follows: 
\[ f(x) = \begin{cases} 3x, & \text{if } x < 0 \\ \min(1+x+\lfloor x \rfloor, 2+x\lfloor x \rfloor), & \text{if } 0 \leq x \leq 2 \\ 5, & \text{if } x > 2 \end{cases} \]
where \(\lfloor . \rfloor\) denotes the greatest integer function. If \(\alpha\) and \(\beta\) are the number of points, where \(f\) is not continuous and is not differentiable, respectively, then \(\alpha + \beta\) equals:

Hint:

When evaluating discontinuity and differentiability for piecewise functions, always check transitions between piecewise segments and integer boundaries within the domain.

Answer 1

Correct Answer: 5

Step 1: Identify discontinuities. The function changes definition at \(x = 0\) and \(x = 2\). Evaluate limits from left and right at these points: \[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} 3x = 0 \] \[ \lim_{x \to 0^+} f(x) = \min(1 + 0 + 0, 2 + 0 \times 0) = 1 \] \[ \lim_{x \to 2^-} f(x) = \min(1 + 2 + 1, 2 + 2 \times 1) = 4 \] \[ \lim_{x \to 2^+} f(x) = 5 \] Discontinuity at \(x = 0\) and \(x = 2\). 

Step 2: Identify points of non-differentiability. Check for differentiability at integer points within \([0, 2]\) and at \(x = 2\), as \(f(x)\) involves the floor function which is non-differentiable at integers: \[ f'(x) \text{ is not defined at } x = 1, 2 \] 

Step 3: Count \(\alpha\) and \(\beta\). \[ \alpha = 2 \text{ (discontinuity at 0 and 2)} \] \[ \beta = 3 \text{ (non-differentiability at 0, 1, and 2)} \] \[ \(\alpha + \beta = 5 \)