GATE CS 2023 Q5 — General Aptitude

To determine the correct statement about a smooth $3$-dimensional object, let's analyze the geometric properties of smooth surfaces:

• Smoothness: A mathematically "smooth" surface is continuously differentiable (typically $C^1$ or $C^2$ smooth). This means the surface has a well-defined tangent plane at every point, which directly implies it cannot have sharp edges or corners. Therefore, Option D is incorrect.
• Curvature (Convexity/Concavity): A smooth object is not restricted to being entirely convex or entirely concave. For example:
  • A sphere is convex everywhere.
  • A dumbbell or a torus (donut shape) has regions that are locally convex (curving outwards) and regions that are locally concave or saddle-like (curving inwards).

Thus, the surface of a smooth $3$-dimensional object may be concave in some places and convex in other places depending on its shape.

Therefore, the correct option is C.