Suppose $f(x)$ is a function that has this property:
For all real numbers $a$ and $b$ such that $a<b$, the portion of the graph of $y=f(x)$ between $x=a$ and $x=b$ lies below the line segment whose endpoints are $(a,f(a))$ and $(b,f(b))$.
(A function with this property is called strictly~convex.)
Given that $f(x)$ passes through $(-2,5)$ and $(2,9)$, what is the range of all possible values for $f(1)$? Express your answer in interval notation.
I have no clue how to start this problem. Any help would be great!
HINT: Let $g$ be a linear fuction passing through $(-2,5)$ and $(2,9)$. Then $f(1)\in(-\infty,g(1))$ (why?).