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The absolute maximum value of $f\left(x\right) = x^3-3x^2+12$ on closed interval $\left[-2,4\right]$ occurs at $x = $

Confused what does absolute maximum value means.

Does it mean

  1. The largest of the large values? $\max \{f\left(x\right)\mid x\in [-2,4]\}$
  2. The largest absolute value of $\max \{\vert f\left(x\right)\vert : x\in [-2,4]\}$

I have figured out that both values are the same when $x=4$, I can see that just from the graph of the function

function

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$\max \{|f(x)| : x\in [-2,4]\}$ = $\max \{f(x): x\in [-2,4]\}$, for this example –  Rustyn Jan 19 '13 at 1:37
    
@RustynYazdanpour That is why I was asking what the term meaned because I couldn't figure which one I should think when I see that colection of words. –  yiyi Jan 19 '13 at 1:45
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absolute maximum means "the largest value" on the interval, @MayoYiyi –  Rustyn Jan 19 '13 at 1:49
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I bet the definition is in your textbook... Some textbooks use terms "relative maximum" and "absolute maximum", while others use "local maximum" and "global maximum". –  GEdgar Jan 19 '13 at 2:48
    
@RustynYazdanpour what edit did you make to my questions, just want to know to improve my questions –  yiyi Jan 21 '13 at 19:31
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2 Answers

up vote 2 down vote accepted

I would guess it is your first option. A usual terminology in calculus is about absolute and relative (or local) maxima and minima.

The absolute maximum would be then $\max\{f(x):\ x\in[-2,4]\}$.

The phrase "absolute maximum value" probably has to do with the fact that when looking at extrema of functions, one usually focus on where they are (i.e. $x=\ldots$) rather than what they are (i.e. $f(x)=\ldots$). The latter is the value, so saying "absolute maximum value" one wants the answer "$28$" as opposed "$x=4$".

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It means your first assumption.

Setting the derivative equal to $0$, we obtain:

$3x^2 - 6x = 0 \Rightarrow$
$x(3x - 6)=0 \Rightarrow$

$x=0,$ or $x=2$

$f(2) = 8$, $f(0) = 12$

Now we test end points,

$f(4) = 64 - 48 + 12 = 28$
$f(-2)= -8 -12 + 12 = -8$

Hence $f(4)=28$ = $\max \{ f(x): x \in [-2,4]\}$

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whoops, i was thinking backwards --Edited-- –  Rustyn Jan 19 '13 at 1:34
    
By the way $\max \{|f(x)| : x\in [-2,4]\}$ = $\max \{f(x) : x\in [-2,4]\}$. in this case –  Rustyn Jan 19 '13 at 1:36
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