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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
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 ¯y(2,12) there must be a value ¯x(2,8) such that:

f(¯x)=¯y

As 7(2,12), then there must be ¯x[2,8] such that:

f(¯x)=7

On the contrary if you let:

f(x)=3x225x+443

f(2)=2

f(8)=12

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

If instead you let:

f(x)=5x43

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