Real Analysis : Intermediate Value Theorem

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• Proof using the Intermediate value theorem

  475627 Proof using intermediate value theorem Let f : [0,1]-->[0,1] be a continuous function. Show that there exists a real number x in [0,1] such that f(x)=x (apply the intermediate value theorem to the function f(x) -x).

• Sets : Intermediate Value Theorem

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• Real Analysis : Continuous Functions

  By the intermediate value theorem, we can find some x in (0,1), such that g(x)=0. This means f(x)-f(x-1)=0, so f(x)=f(x-1). Case 3: g(0)<0, then g(1)=-g(0)>0. Also by the intermediate value theorem, we can find some x in (0,1), such that g(x)=0.

• Polynomial Functions and Intermediate Value Theorem

  544489 Polynomial Functions and Intermediate Value Theorem 1. Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.

• Polynomials and Rational Theorem Algebra Questions

  Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.

• Properties of Continuous Functions : Intermediate Value Theorem

  56412 Properties of Continuous Functions : Intermediate Value Theorem Suppose that functions f,g : [a,b] -> R are continuous, satisfy f(a) <= g(a) and f(b) >= g(b). Then there exists a real number c in [a,b] such that f(c) = g(c).

• If f is a function from R to R, and there exists a real number aE(0,1) such that |f'(x)|≤a for all xER , show that the equation x = f(x) has a solution.

  To see this, we can consider the difference By (1), and the intermediate value theorem, we have ........................(2) where is between and .