Suppose f : [ − 1 , 1 ] → R is continuous, f ( − 1 ) &gt; − 1, and f ( 1 ) &lt; 1. Show that f has a fixed point.So for every y between f ( − 1 ) and f ( 1 ) we can find an x s.t. − 1 &lt; x &lt; 1 and f ( x ) = y, but I'm not sure how to show one of these creates a fixed point.

Question

Suppose   f  :  [  −  1  ,  1  ]  →      R   is continuous,   f  (  −  1  )  &amp;gt;  −  1, and   f  (  1  )  &amp;lt;  1. Show that   f has a fixed point.So for every   y between   f  (  −  1  ) and   f  (  1  ) we can find an   x s.t.   −  1  &amp;lt;  x  &amp;lt;  1 and   f  (  x  )  =  y, but I&#039;m not sure how to show one of these creates a fixed point.

Gaige Burton · Accepted Answer

Try the usual trick. Let   g  (  x  )  =  x  −  f  (  x  ). Since   f  (  x  ) is continuous,   g  (  x  ) is also continuous. Now   g  (  −  1  )  &amp;lt;  0  &amp;lt;  g  (  1  ). Applying the intermediate value theorem, there must be a   c  ∈  (  −  1  ,  1  ) such that   g  (  c  )  =  0. Hence,   g  (  c  )  =  c  −  f  (  c  )  =  0

Lokubovumn · Accepted Answer

So   g takes on all values between   g  (  −  1  )  &amp;lt;  0 and   g  (  1  )  &amp;gt;  0, hence there must be some point at which it equals zero.

Suppose f:[−1,1]->R is continuous, f(−1)>−1, and f(1)<1. Show that f has a fixed point. So for every y between f(−1) and f(1) we can find an x s.t. −1<x<1 and f(x)=y, but I'm not sure how to show one of these creates a fixed point.

Answered question

Answer & Explanation

New Questions in Calculus 2