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Functions in Normed Linear Space 
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#37
Dec1311, 02:47 PM

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Sorry, ignore this post. Thanks 


#38
Dec1311, 02:53 PM

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a <= 6 is equivalent to 6 <= a <= 6.



#39
Dec1311, 03:10 PM

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How would you verify whether another new function like z(x)=5x is an element in A? According to Wolfram, it looks like this function is with the region... http://www.wolframalpha.com/input/?i...etween+0+and+1 


#40
Dec1311, 04:37 PM

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PF Gold
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[tex]\fz\=\sup_{t\in[0,1]}f(t)z(t)=\sup_{t\in[0,1]}t^3+35t=3[/tex] The last step is easy since the derivative of fz doesn't have any zeroes in [0,1]. This means that fz is either increasing in the entire interval, or decreasing in the entire interval. Since (fz)(0)=3 and (fz)(1)=1, it must be decreasing, and the maximum value is 3.



#41
Dec1311, 05:19 PM

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Is knowing what the resulting max value not enough, to determine whether fz is in A? Thanks Fredrik :) 


#42
Dec1311, 05:32 PM

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What you need to determine is the max value in the interval. Since the slope is never zero inside the interval, we know that the function has its max value at one of the endpoints of the interval.
If the slope had been zero somewhere in the interval, then that might have been the point where the function has its max value. 


#43
Dec1311, 06:00 PM

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If a differentiable function f is increasing on an interval [a, b], its smallest value is f(a) and its largest is f(b). OTOH, if f is decreasing on [a, b], its largest value is f(a) and its smallest is f(b). These are very simple ideas. 


#44
Dec1411, 02:18 PM

P: 652

I am just looking for an example that demonstrates the slope being 0 somewhere in the interval instead of at the ends.... You put in 0 and you are left with 3. If you put in 1 we get 1. So we take the highest of those two because [itex]sup_{t \in [0,1]}  fz=max\{t_1,t_2\}[/itex] or something like this? 


#45
Dec1411, 03:07 PM

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#46
Dec1411, 03:38 PM

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#47
Dec1411, 04:24 PM

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#48
Dec1511, 02:14 AM

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#49
Dec1511, 02:52 AM

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#50
Dec1511, 03:12 AM

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#51
Dec1511, 12:56 PM

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#52
Dec1511, 01:12 PM

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PF Gold
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I have been reading this thread since my first post which was #2. I am curious bugatti79 about two questions:
1. What course are you currently studying where this topic arose? 2. Have you taken a course in calculus? 


#53
Dec1511, 01:35 PM

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It looks like you need to refresh your memory about some basic calculus (as a couple of people have already suggested). 


#54
Dec1511, 01:55 PM

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2) No I have not taken a course in calculus but I come from an engineering background so I have some exposure to basic calculus which I guess I need to brush up on!! I am finish with thread now 


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