Looking at the surface of a smooth 3-dimensional object from the outside,…
2023
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.
Attempted by 73 students.
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Correct answer: C
Answer: The surface of the object may be concave in some places and convex in other places.
Explanation: A smooth surface means it is differentiable everywhere (no sharp corners or edges), but the curvature at each point can vary. Locally the surface can curve outward (locally convex), inward in some directions (locally concave or saddle-shaped), or be flat.
Why the surface does not have to be concave everywhere: 'Concave everywhere' would mean every point curves inward, which is not required; many smooth shapes curve outward in regions.
Why the surface does not have to be convex everywhere: 'Convex everywhere' would forbid inward-curving parts; smooth objects like a torus or a mug can have both inward and outward curvatures.
Why the statement about edges but no corners is incorrect: sharp edges and corners are non-smooth features. A smooth surface has a well-defined tangent plane at every point, so it cannot have true sharp edges or corners.
Examples: a sphere is convex at every point; a torus has regions of both positive and negative curvature; many real objects (for example a cup or parts of the human body) show mixed concave and convex regions.