I need help w/ "Find the infimum of G = { x | (x​^2 ≤ 7 }"

[allos]

1. Find the infimum of
G={x|(x^​2≤7}

solve the inequality h[FONT=MathJax_Main]([/FONT]x[FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT]
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1. [FONT=MathJax_Math-italic]x​
   [FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]12[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT]
   
   
   ​
   [/FONT]

[FONT=MathJax_Math-italic]G​

[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Size3]{[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT]
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[Ishuda]

1. Find the infimum of

[FONT=MathJax_Math-italic]G[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Size1]{[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Size1]}[/FONT][/FONT]
solve the inequality h[FONT=MathJax_Main]([/FONT]x[FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT]
​

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1. [FONT=MathJax_Math-italic]x​
   [FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]12[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT]
   
   
   ​
   [/FONT]

[FONT=MathJax_Math-italic]G​

[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Size3]{[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT]
​

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What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

HINTS: For the first problem (and I assume the x² is supossed to be an x2), what does infimum mean? If x were -2 would x² still be less than 7?

For the second problem: If the expression is what as I think
x² + x + 1 x² - 7x + 12
first collect terms. Then recognize that it is a quadratic so it has either 0, 1, or, 2 zeros and the using quadratic formula will tell you which is true and provide the zeros if there are any. Then look at the coefficient in front of the x2. If it is positive, the function becomes very large for very large positive and very large negative x and that should be enough to determine where the expression is positive. If the coefficient is negative, what happens for very large positive x and very large negative x? Again, that, along with the zero's should be enough to determine where the expression is positive. Alternatively, you can graph the expression to determine where the expression is positive [use the quadratic formula to find the exact zero's if the graph doesn't provide them].

 

[pka]

1. Find the infimum of G={x|(x^​2≤7}

This is a very important concept to understand if you need to work beyond very basic calculus.
There is a difference is infimum, \(\displaystyle \inf(G)\) and the minimum, \(\displaystyle \min(G)\).

The \(\displaystyle \min(G)\) may not exist, but if it does \(\displaystyle \min(G)\in G\).

If a set if bounded below, as in this case, then \(\displaystyle \inf(G)=-\sqrt7\) exists and is in \(\displaystyle G\).