Quasi concavity and quasi convexity of a max function

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Consider the production function: Q = max {K, L}. We have to find out whether this function is quasi concave or quasi convex.

According to me, since this function is concave, it should automatically be quasi concave (also the upper contour sets are convex). And It is quasi convex because the lower contour sets are convex.

However my answer book states that these are only quasi convex and NOT quasi concave. I dont understand why?