How is intermediate value theorem valid for sin ⁡ x in [ 0 , π...

letumsnemesislh

letumsnemesislh

Answered

2022-07-08

How is intermediate value theorem valid for sin x in [ 0 , π ]?

It has max value 1 in the interval [ 0 , π ] which doesn't lie between values given by sin 0 and sin π.

Answer & Explanation

trantegisis

trantegisis

Expert

2022-07-09Added 20 answers

Let f : [ a , b ] R be a continuous function and let y R . The intermediate value theorem says that if f ( a ) y f ( b ) or if f ( a ) y f ( b ), then there is a c [ a , b ] such that f ( c ) = y. But it says nothing if y lies outside the interval bounded by f ( a ) and f ( b ). So, there is no contradiction here.
tripes3h

tripes3h

Expert

2022-07-10Added 5 answers

That's not how it works. The Intermediate Value Theorem says that if f is continuous on [ a, b], then it achieves every value between
c = min { f ( x ) :   x [ a , b ] } ,
and
d = max { f ( x ) :   x [ a , b ] } .
When f is monotone, it happens that c , d are f ( a ) , f ( b ), but in general it is not the case.

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