Let f be a continuous function on the closed interval [ − 2 , 5 ]. If f((-2)=-7 and f(5)=1, then the Intermediate Value Theorem guarantees t

Question

Let f be a continuous function on the closed interval [ − 2 , 5 ]. If f((-2)=-7 and f(5)=1, then the Intermediate Value Theorem guarantees that

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Remi 2 weeks 2021-11-17T16:44:48+00:00 1 Answer 0 views 0

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    2021-11-17T16:45:52+00:00

    Answer:

    The function has at least 1 zero within the interval [-2,5].

    Step-by-step explanation:

    The intermediate value theorem states that, for a function continuous in a certain interval [a,b], then the function takes any value between f(a) and f(b) at some point within that interval.

    This theorem has an important consequence:

    If a function f(x) is continuous in an interval [a,b], and the sign of the function changes at the extreme points of the interval:

    f(a)>0\\f(b)<0 (or viceversa)

    Then the function f(x) has at least one zero within the interval [a,b].

    We can apply the theorem to this case. In fact, here we have a function f(x) continuous within the interval

    [-2,5]

    And we also know that the function changes sign at the extreme points of the interval:

    f(-2)=-7<0\\f(5)=1>0

    Therefore, the function has at least 1 zero within the interval [-2,5], so there is at least one point x’ within this interval such that

    f(x')=0

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45:7+7-4:2-5:5*4+35:2 =? ( )