Answer to Question #244077 in Calculus for Princewill Unuigbe

Question

 For the function f(x) = x 2 use the Intermediate Value Theorem to prove that there exists a number c such that f(c) = 2. Your are proving the existence of the real number √ 2. Since f(1) < 2 < f(2) you can restrict f(x) to the interval [1, 2]

Answer 1

$Here$ $f(x)=x^2$ .It is defined on R .Let us consider an interval [1,2]

Here $f(1)=(1)^2=1$ , $f(2)=(2)^2=4$

According to intermediate value theorem there exists an element k satisfying f(a)<k<f(b) then f(c)=k where c belongs to (a , b) . Here 2 $\epsilon$ (1,4) .So f(c)=2 $\implies$ $c^2=2\implies$ $c=\sqrt2$ (proved)