2) Let F(x) be a continuously differentiable function defined on the interval (a, b) such that F(a) <0, F(b) > 0 and
0 < K1 < F' (x) < K2 (a < x < b)
Use Theorem 1 to find the unique root of equation F(x) - 0.
Hint. Introduce the auxiliary function f(x) = x - λF(x), and choose λ such that the theorem works for the equivalent equation f(x) = x.
THEOREM Given a continuous mapping of a complete metric space R into inself, suppose A" is a contraction mapping (n an integer > 1). Then A has a unique fixed point.
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