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Deduce properties of the solutions of a given differential equation

Let f(x) be a solution for all x \in \mathbb{R} of the second order differential equation

    \[ xf''(x) + 3x(f'(x))^2 = 1 - e^{-x}. \]

Without attempting to solve the equation answer the following questions.

  1. If c \neq 0 and f has an extremum at c, prove that this extremum must be a minimum.
  2. If f has an extremum at 0 decide whether this extremum is a maximum or minimum and prove your assertion.
  3. If the solution f(x) satisfies the conditions

        \[ f(0) = f'(0) = 0 \]

    find the minimum value for the constant A such that

        \[ f(x) \leq Ax^2 \qquad \text{for all } x \geq 0. \]


Incomplete.

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