We are provided with two convex functions f(x) and g(x) and we are supposed to comment on the convexity of min{f(x),g(x)}.
I start out by writing the equations implied when we convexity of f(x) and g(x) are stated,
i.e f[λx + (1-λ)y] <= λf(x) + (1-λ)f(y)....(i)
and, g[λx + (1-λ)y] <= λg(x) + (1-λ)g(y)....(ii)
Now let us assume another function, say h(x)= min{f(x),g(x)}
Since h(x)= min{f(x),g(x)} h(x) <= f(x) and h(x) <= g(x)
Now since h(x) <= f(x)
Cannot we write equation (i) as h[λx + (1-λ)y] <= λf(x) + (1-λ)f(y)
Similarly, equation (ii) as h[λx + (1-λ)y] <= λg(x) + (1-λ)g(y)
and with these two new equations we are able to conclude that
h[λx + (1-λ)y] <= min[λf(x) + (1-λ)f(y),λg(x) + (1-λ)g(y)]
=> h[λx + (1-λ)y] <= λmin[f(x),g(x)] + (1-λ)min[f(y),f(y)]
=> h[λx + (1-λ)y] <= λh(x) + (1-λ)h(y)
Which proves h(x) which is min{f(x),g(x)} is a convex function but searching many websites including math.stackexchange I got to know that h(x) is not necessarily convex so where have I gone wrong? Please help!