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What, if anything, does the Mean Value Theorem gaurantee for the given function on the given interval?

f(x)=x2-2x+5  on [1,4]

 Oct 16, 2016
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f(x)  =  x2 - 2x - 5

When x = 1:  f(1))  =  12 - 2(1) - 5  =  1 - 2 - 5  =  -6   --->  one endpoint is (1, -6).

When x = 4:  f(4))  =  42 - 2(4) - 5  =  16 - 8 - 5  =  3   --->  one endpoint is (4, 3).

 

The line through the endpoints  (1, -6)  and  (4, 3)  has slope:  m  =  (3 - -6) / (4 - 1)  =  9/3  =  3.

 

The Mean Value Theorem states that there is at least one point on the function  f(x)  =  x2 - 2x - 5  in the open interval  (1, 4)  that has the same slope (m = 3).

 Oct 16, 2016

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