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

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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naseeamAsked:
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ozoCommented:
an extrema of a continuous function is either where the derivative changes sign or at the endpoints of the domain
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d-glitchCommented:
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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phoffricCommented:
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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naseeamAuthor Commented:
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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phoffricCommented:
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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HainKurtSr. System AnalystCommented:
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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ozoCommented:
each partition having no absolute value functions
or more generally, each partition being monotonic.
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