Suppose $f(x)$ is a function that has this property:

For all real numbers \(a\) and \(b\) such that $a \(y=f(x)\) 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.

Guest May 11, 2018

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Guest
May 11, 2018

edited by Guest May 11, 2018

edited by Guest May 11, 2018

edited by Guest May 11, 2018

edited by Guest May 11, 2018

edited by Guest May 11, 2018

edited by Guest May 11, 2018

#4**+2 **

Let a=-2 and b=2. Then the special property of f(x) tells us that the point (1,f(1)) lies below the line segment through (-2,5) and (2,9). This line segment is part of the line with equation y=x+7, so it passes through (1,8). Therefore, f(1)<8. There are no further restrictions on f(1). Thus, as an interval, the range of possible values for f(1) is \(\boxed{(-\infty,8)}\)

Guest May 12, 2018