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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
edited by Guest  May 11, 2018
edited by Guest  May 11, 2018
edited by Guest  May 11, 2018
edited by Guest  May 11, 2018
 #1
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I think that a part of your question is missing.

Melody  May 11, 2018
 #2
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The whole second part of the second sentence wouldn't load onto the screen, until I replaced some of the variables with LATEX. sry

Guest May 11, 2018
 #4
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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
 #5
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Yes I think guest is correct, thanks guest :)

Melody  May 12, 2018

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