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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!

TheLeogend
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  • Start with: given a straight line passing through the given points, what is the $y$ value on that line at $x=1$? – Joffan May 08 '18 at 22:20

1 Answers1

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HINT: Let $g$ be a linear fuction passing through $(-2,5)$ and $(2,9)$. Then $f(1)\in(-\infty,g(1))$ (why?).

Przemysław Scherwentke
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