Four analytic functions are defined for all real

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Four analytic functions are defined for all real xx. u(x)=3x2−3u(x) = 3x^{2} - 3 v(x)=2sin⁡xv(x) = 2\sin x w(x)=x3−3x+1w(x) = x^{3} - 3x + 1 z(x)=6xz(x) = 6x Exactly three of these functions constitute the graphs of a differentiable function ff together with its first and second derivatives. The remaining function does not belong to that set. (1) Part 1: Identify which functions represent ff, f′f' and f′′f'', and indicate which function is extraneous. Provide complete justification, including at least one derivative computation or comparison for each assignment. (2) Part 2: For the function you have designated as ff, determine all intervals of concavity and locate every point of inflection. (3) Part 3: Explain why the function you declared extraneous cannot represent any of ff, f′f' or f′′f'', even up to multiplication by a non-zero constant factor.

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