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Let f : R → R be defined by f(x) = x^3 + 3x, for all...

Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i) Prove that if y > 0, then there is a solution x to the equation f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove that the function f : R → R is strictly monotone. (iii) By (i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why the derivative of the inverse function, (f^ −1 )'(y) exists, for all y ∈ R, and find its value at y = 0.

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