Given: the convex polygon with 17 sides. Is it possible to cut it into 14 triangles?

No. The first triangle is formed from three ordered vertices v0, v1, and v2. The other 14 vertices v3, v4, ..., v16 are not part of this first triangle. Thereafter, each new vertex adds one additional triangle in fan or strip topology so there are 14 additional triangles. The total is 15.