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Let \(f\) be the exponential function defined on \(\mathbb{R}\) by \(f(x) = e^x\).
  1. Justify that \(f\) is strictly convex on \(\mathbb{R}\).
  2. Determine the equation of the tangent \(T\) to the graph of \(f\) at the point with abscissa \(0\).
  3. Using the properties of convex functions, prove that for all \(x \in \mathbb{R}\), \(e^x \ge x + 1\).

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