f is continuous function in [0,2] and f(2)=1 proof that there is a point x in [0,2] so that: f(x)=1/x

pumicayf

pumicayf

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2022-08-22

f is continuous function in [ 0 , 2 ] and f ( 2 ) = 1
proof that there is a point x in [ 0 , 2 ] so that:
f ( x ) = 1 x
nobody in my class solved this exercise. our teacher give us a hint: p ( x ) = x f ( x ) and to do Intermediate value theorem on this function. so I tried:
p ( 0 ) = 0 f ( 0 ) = 0 , P ( 2 ) = 2 f ( 2 ) = 2
p ( 0 ) < h < p ( 2 ) by Intermediate value theorem, there is a point c so that a < c < b and f ( c ) = h
please solve

Answer & Explanation

saniraore

saniraore

Beginner2022-08-23Added 7 answers

You seem to be almost done. You know that p ( 0 ) = 0, p ( 2 ) = 2 and that p is continuous. You want to find an x [ 0 , 2 ] such that f ( x ) = 1 / x, in other words p ( x ) = 1. Thus you can set h = 1 and you know there exists a c [ 0 , 2 ] such that p ( c ) = 1. This in turn means f ( c ) = 1 / c, so you're done.
sponsorjewk

sponsorjewk

Beginner2022-08-24Added 2 answers

Hint: If p ( x ) = h, then
f ( x ) = h x
You're free to choose h to be anything between 0 and 2, and there's a particularly salient choice of h.

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