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The book states that $\sup_{x \in D} f(x) = \sup_{-1\le x \le5}(x^2 - 9x +1 ) = 11$

And $\inf_{x \in D} f(x) = \inf_{-1\le x \le5}(x^2 - 9x +1 ) = \frac{-77}{4}$

my question is that why $\sup f(x) = 11$ and $\inf f(x) = \frac {-77}{4}$

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    Draw the graph of the function, and you will see it. Can you draw the parabola? – 2017-02-28
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    I did from the interent, but I thoguth sup f(x) means that biggest number y and inf f(x) means that smallest number y. But it doesn't seem like it. – 2017-02-28

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Hint. Note that $$f(x)=x^2-9x+1=\left(x-\frac{9}{2}\right)^2-\frac{81}{4}+1=\left(x-\frac{9}{2}\right)^2-\frac{77}{4}.$$ This implies that $f$ is decreasing in $[-1,9/2]$an it is increasing in $[9/2,5]$.

Can you take it from here?

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    um.. I don't actually get it.. – 2017-02-28
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    Does $\sup f(x)$ mean that biggest number $y$ between $-1< x< 5$ ? and inf f(x) means that smallest number $y $ between $-1 < x< 5$ ? – 2017-02-28
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    Yes, something like that. So, by using my hint, you have that the supremum is $f(-1)$ or $f(5)$ (check the values) and the infimum is $f(9/2)=-77/4$. – 2017-02-28
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    But how can I find biggest number y and smallest number without drawing, graph.. ? I know that $x^2 - 9 x + 1 $ is U shape graphe. so, I can find the smallest number from $2x - 9 = 0$ . But how about biggest number y..? I can guess that since graph has U shape, biggest number should be left (-1) or right (5) side of graph. Is this right ? I took calculous class two years ago, so I am not sure whether this is right or not.. – 2017-02-28
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    Yes, you are right! – 2017-02-28