Differential Calculus-1
Differential Calculus-1
Allen Star Batch
Grade 12
Question:
The function $f(x) = [x] + \sqrt{\{x\}}$, where $[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function respectively, is discontinuous at
all $x$
all integer points
no $x$
$x$ which is not an integer
Step-by-Step Solution
Key Concept: Graphical representation shows monotonically increasing behavior with increasing rate of change.
The graph of $f(x)$ shown illustrates a function that increases monotonically from the left, passing through points at $x = 1, 2, 3$. The curve exhibits increasing slope, suggesting the derivative increases as $x$ increases, consistent with a convex function.
Correct Answer: 3