Math 2.0

statement proposition

The exponential function is convex

Statement

The exponential function is strictly convex on R\mathbb{R}. That is, etx+(1−t)y<tex+(1−t)eye^{t x + (1 - t) y} < t e^x + (1 - t) e^y for all real x≠yx \ne y and t∈(0,1)t \in (0, 1).

Remarks

The second derivative of exe^x is exe^x itself, which is positive. So the second derivative test for convexity applies, in its strict form. The tangent line at 00 gives the inequality 1+x≤ex1 + x \le e^x. Mathlib states convexity as convexOn_exp.

Explanations

L1 human

The graph of exe^x bends upward everywhere. Every chord of the graph lies above the graph.

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What this needs

To read this node, read these 54 nodes in this order. The foundations come first, and the list ends with this node.

  1. Convex function definition
  2. Limit point of a set definition
  3. Local maximum definition
  4. Sequence definition
  5. Slopes of a convex function are increasing statement
  6. Limit of a function definition
  7. Local minimum definition
  8. Real sequence definition
  9. Subsequence definition
  10. Continuity at a point definition
  11. Derivative definition
  12. Bounded sequence definition
  13. Geometric sequence definition
  14. Limit of a sequence definition
  15. Monotone sequence definition
  16. Series definition
  17. Continuous function definition
  18. Differentiable function definition
  19. Fermat's theorem on stationary points statement
  20. Sum of a finite geometric series statement
  21. Monotone convergence theorem for sequences statement
  22. Convergent sequence definition
  23. Squeeze theorem statement
  24. Every sequence has a monotone subsequence statement
  25. Cauchy product definition
  26. Partial sum definition
  27. Power series definition
  28. Derivative function definition
  29. The sequence r^n tends to zero for |r| < 1 example
  30. Bolzano–Weierstrass theorem statement
  31. Radius of convergence definition
  32. Convergent series definition
  33. Higher derivative definition
  34. Boundedness theorem statement
  35. Absolute convergence definition
  36. A series of nonnegative terms converges iff its partial sums are bounded statement
  37. Sum of a series definition
  38. Second derivative definition
  39. Extreme value theorem statement
  40. Geometric series example
  41. Comparison test statement
  42. Cauchy product of absolutely convergent series statement
  43. Rolle's theorem statement
  44. Ratio test statement
  45. Mean value theorem statement
  46. Ratio formula for the radius of convergence statement
  47. A differentiable function is convex iff its derivative is increasing statement
  48. A function with nonnegative derivative is increasing statement
  49. The exponential series has infinite radius of convergence statement
  50. A function with nonnegative second derivative is convex statement
  51. Exponential function definition
  52. Exponential of a sum statement
  53. The exponential function is its own derivative statement
  54. The exponential function is convex statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Properties

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