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SOLVED:Let f be a continuous real-valued function on [0,1]. Prove that given ϵ>0, there exists a polynomial P with rational coefficients such that |f(x)-P(x)|<ϵfor all x ∈[0,1].

SOLVED:Let f be a continuous real-valued function on [0,1]. Prove tha…

(c) Let f be a real-valued continuously differentiable function mapping the closed interval [a, b] to itself. Suppose there is a constant 0 < M < 1 such that |f'(x)| < M for each x âˆˆ [a, b]. Show that f has a unique fixed point.Let f be a real-valued continuously differentiable function defined on the closed interval [a, b]. Suppose that f(a) and f(b) have opposite signs. Suppose in addition that 0 < M1 < |f'(x)| < M2 for each x âˆˆ [a, b]. Show that f has a unique root in [a, b]. (Hint: For a

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