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How to find absolute extrema of  4 - abs(t - 4)

Posted on 2014-11-22
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Last Modified: 2014-11-24
Determine the absolute extrema of the function and the x-value in the closed interval where it occurs.

h ( t )  =  4  -   abs ( t - 4 )      
closed interval is  [ 1, 6 ]

This is not a homework problem.  

This problem is from high school Advance Placement Calculus class.
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Question by:naseeam
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8 Comments
 
LVL 84

Assisted Solution

by:ozo
ozo earned 50 total points
ID: 40460203
an extrema of a continuous function is either where the derivative changes sign or at the endpoints of the domain
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Assisted Solution

by:d-glitch
d-glitch earned 100 total points
ID: 40460215
You should plot the function.  Try the integers in the range of interest.
Do you know what the plot of abs(t) looks like?  It is always GTE zero.
So what does 4 abs( ) look like?
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LVL 32

Accepted Solution

by:
phoffric earned 350 total points
ID: 40460283
I would break the closed interval of [1, 6 ] into two partitions: one where t - 4 >= 0 and one where t - 4 < 0. Now you have two different functions h1(t) and h2(t), one for each partition with no abs() function involved. Removing the abs() function usually makes these problems easier to understand. Analyze absolute extrema on both partitions, which should be easy.

I assume that you mean "the t-value" rather than "the x-value".
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by:naseeam
ID: 40460836
h1(t) = 4 - ( t - 4 ) ;        [4, 6 ]
h2(t) = 4 - ( - t  +  4);     [1, 4 ]

h1 ' ( t )  =  - 1
h2 ' ( t ) =  1

At t = 4, slope is undefined.  There is absolute maximum at t = 4.

Am I correct ?
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LVL 32

Expert Comment

by:phoffric
ID: 40460854
I hoped you first simplified h1(t) and h2(t).
You are correct.

In general, as you progress into harder functions and inequality relationships with multiple absolute value functions, the task first becomes identifying multiple partitions and then defining functions on each partition having no absolute value functions. A bit harder than this problem if the individual regions for different absolute value expressions overlap.
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LVL 56

Expert Comment

by:HainKurt
ID: 40461224
strange.. I remember I posted something here, but I dont see now... did someone delete my post?

I dont see my post but i still get notifications? what happened to my post (and possible points)?
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LVL 84

Expert Comment

by:ozo
ID: 40461588
each partition having no absolute value functions
or more generally, each partition being monotonic.
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