Interval related to increasing/decreasing and concavity/convexityWhy do some people use closed intervals when describing the intervals where a function is increasing/decreasing or concave/convex?For example, given the function f ( x ) = x 2 − 5 x + 6, it says the interval of increase is [ 5 / 2 , ∞ ). Why is this written as a closed interval, and not an open one?Concavity, on the other hand, uses open intervals.

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Interval related to increasing/decreasing and concavity/convexityWhy do some people use closed intervals when describing the intervals where a function is increasing/decreasing or concave/convex?For example, given the function   f  (  x  )  =      x    2    −  5  x  +  6, it says the interval of increase is   [  5      /    2  ,  ∞  ). Why is this written as a closed interval, and not an open one?Concavity, on the other hand, uses open intervals.

falwsay · Accepted Answer

Explanation:In an open interval the domain is unbounded when concavity is considered. The particular example function is symmetrical about   x  =      5    2   so includes full range   +  ∞  ,  −  ∞  ..

shadyufog0 · Accepted Answer

Step 1Increasing or decreasing compares the function value at 2 points in the interval.If       f    ′    &amp;gt;  0 on (a,b) and f is continuous on [a,b] then if x is such that   a  &amp;lt;  x  &amp;lt;  b then   f  (  x  )  &amp;lt;  f  (  b  ) so f IS increasing on [a,b].Step 2On the other hand, concavity is generally an attribute at a specific point, so if       f    ″    &amp;gt;  0 for   x  &amp;lt;  a and       f    ″    &amp;lt;  0 for   x  &amp;gt;  a, we wouldn&#039;t say f is concave up or down at   x  =  a.The reason I put the modifier &quot;generally&quot; in there is in the case where       f    ″    (  a  )  =  0 but       f    ″    &amp;gt;  0 for   x  &amp;lt;  a and       f    ″    &amp;lt;  0 for   x  &amp;gt;  a. Then we still say f is concave up at a. (Think of   f  (  x  )  =      x    4  .)

Given the function f(x)=x^2-5x+6, it says the interval of increase is [5/2, infty). Why is this written as a closed interval, and not an open one?

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