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the f be a function that is continuous on the closed interval [2,8]. Given that f(2)=2 and f(8)=12, which of the following statements is guaranteed by the intermediate value theorem

  1. f(x) is linear
    2.f(x) is always increaseing
    3.f(x)=7 has at least one solution in the open interval (2,8)
    4.f(5)=7
    5.there is at least one maximum on the open interval (2,8)

1 Answer
Andrea S.
Jun 28, 2018

The third one.

Explanation:

The third one.

In fact as #f(x)# is continuous in #[2,8]#, and #f(2) =2#, #f(8) = 12#, the intermediate value theorem states that for every #bar y in (2,12)# there must be a value #barx in (2,8)# such that:

#f(bar x) = bary#

As #7 in (2,12)#, then there must be #barx in [2,8]# such that:

#f(barx) = 7#

On the contrary if you let:

#f(x) = (3x^2 -25x+44)/3 #

#f(2) = 2#

#f(8) = 12#

the function is continuous for #x in [2,8]# but is not linear, it's not strictly increasing and #f(5) !=7#.

If instead you let:

#f(x) = (5x-4)/3#

then the hypotheses are still satisfied but #f(x)#has no maximum in #(2,8)#

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