Looking at the surface of a smooth 3-dimensional object from the outside, which one of the following options is TRUE? A. The surface of the object must be concave everywhere. B. The surface of the object must be convex everywhere. C. The surface of the object may be concave in some places and convex in other places. D. The object can have edges, but no corners.

GATE 2023 · General Aptitude · Patterns In Three Dimensions · medium

Answer: C. The surface of the object may be concave in some places and convex in other places.

1. Eliminate Option A (all concave): A sphere is smooth and entirely convex. Therefore it is false that a smooth 3D object MUST be concave everywhere. Option A is FALSE.
2. Eliminate Option B (all convex): A torus (donut shape) is smooth but its inner ring region (viewed from outside) is concave. Therefore a smooth object need not be convex everywhere. Option B is FALSE.
3. Evaluate Option C (mixed curvature allowed): The torus example confirms that a single smooth 3D object can have concave regions (inner hole) and convex regions (outer surface). There is no geometric law preventing this. Option C is TRUE.
4. Evaluate Option D (edges but no corners): The problem specifies a SMOOTH object. A smooth surface by definition has no sharp edges or corners anywhere. Option D is inconsistent with the smooth constraint. Option D is FALSE.