What are Convex and Concave Curves?

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Let us, first, get clear about naming.

Convex is also called Concave Upward or Convex Downward.
Concave is also called Concave Downward or Convex Upward. 

We call a curve Convex when the slope increases.
We call a curve Concave when the slope decreases.
Hope, the image below will clarify thoroughly.

To get rid of any confusion about naming, remember that when we drop the term 'downward' from any type of curve it reduces to its proper name.
For example, Concave Downward = Concave.
Some examples:

Grateful Mention: Math is Fun