Is linear function convex or concave?
Solution 1:
A linear function is both. Use this definition of convexity:
For any two points $x_1$ and $x_2$ $$\forall a \in [0,1] \quad f(ax_1 + (1-a)x_2) \leq af(x_1) + (1-a)f(x_2)$$
Flip inequality for concave. Do you see why linear is both?
Solution 2:
Linear function is both convex and concave. You may be interested in this page.