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.

Question

How is intermediate value theorem valid for   sin  ⁡  x in   [  0  ,  π  ]?It has max value 1 in the interval   [  0  ,  π  ] which doesn&#039;t lie between values given by   sin  ⁡  0 and   sin  ⁡  π.

trantegisis · Accepted Answer

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 · Accepted Answer

That&#039;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.