A number a is called a fixed point of a function f if f ( a ) = a. Consider the function f ( x ) = x 87 + 4 x + 2, x ∈ R .(a) Use the Mean Value Theorem to show that f ( x ) cannot have more than one fixed point.(b) Use the Intermediate Value Theorem and the result in (a) to show that f ( x ) has exactly one fixed point.

Question

A number   a is called a fixed point of a function   f if   f  (  a  )  =  a. Consider the function   f  (  x  )  =      x          87        +  4  x  +  2,   x  ∈      R  .(a) Use the Mean Value Theorem to show that   f  (  x  ) cannot have more than one fixed point.(b) Use the Intermediate Value Theorem and the result in (a) to show that   f  (  x  ) has exactly one fixed point.

Ashlynn Delacruz · Accepted Answer

(a) If   f has two distinc fixed points, namely   a  &amp;lt;  b, then  f  (  b  )  −  f  (  a  )  =      f    ′    (  c  )  (  b  −  a  )for some   c  ∈  (  a  ,  b  ). Then       f    ′    (  c  )  =  1. But       f    ′    (  x  )  =      x          86        +  4  ≥  4.(b)Let   F  (  x  )  =  f  (  x  )  −  x, which is continuous.   F  (  0  )  =  2 and   F  (  −  1  )  =  −  2. So there is   c  ∈  (  −  1  ,  0  ) such that   F  (  c  )  =  0 and, hence,   f  (  c  )  =  c. By (a), this is the only possible fixed point.

gemauert79 · Accepted Answer

f  (  x  )  −  x is an increasing function. It is positive at   x  =  0 and negative at   x  =  −  1.

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