Suppose that f is a continuous function and that f ( − 1 ) = f ( 1 ) = 0. Show that there is c ∈ ( − 1 , 1 ) such that f ( c ) = c 1 − c 2 I am not sure if this is a new question as I set it this morning, after solved a similar question. I wanted to prove it using the same idea (Intermediate Value Theorem) but it was not that nice...Can anyone help me on this?By the way, the 'similar question' I mentioned above is as follow:Suppose that f is a continuous function and that f ( 0 ) = 1 and f ( 1 ) = 2. Show that there is c ∈ ( 0 , 1 ) such that f ( c ) = 1 c The hint given was to let g ( x ) = x f ( x ) and use the Intermediate Value Theorem.

Question

Suppose that   f is a continuous function and that   f  (  −  1  )  =  f  (  1  )  =  0. Show that there is   c  ∈  (  −  1  ,  1  ) such that  f  (  c  )  =      c          1      −              c        2            I am not sure if this is a new question as I set it this morning, after solved a similar question. I wanted to prove it using the same idea (Intermediate Value Theorem) but it was not that nice...Can anyone help me on this?By the way, the &#039;similar question&#039; I mentioned above is as follow:Suppose that   f is a continuous function and that   f  (  0  )  =  1 and   f  (  1  )  =  2. Show that there is   c  ∈  (  0  ,  1  ) such that  f  (  c  )  =      1    c  The hint given was to let   g  (  x  )  =  x  f  (  x  ) and use the Intermediate Value Theorem.

Kharroubip9ej0 · Accepted Answer

Let   g  (  x  )  =  [  f  (  x  )  (  1  −      x    2    )  ]  −  x. Then   g  (  −  1  )  =  1 and   g  (  1  )  =  −  1.

Suppose that f is a continuous function and that f ( − 1 )

Answered question

Answer & Explanation

New Questions in Calculus 2