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$$f:[0,\infty )\rightarrow \mathbb{R}\textrm{ continuous and }f(0)=0. \, \, f\textrm{ is differentiable on }(0,\infty )\textrm{ and }\left | f'(x) \right |\leq \left | f(x) \right |\textrm{ for all }x> 0. \textrm{ Prove that }\left | f(x) \right |\leq e^{x}\textrm{ for all }x\in \mathbb{R^{+}}$$

Have tried standard theorems taught in analysis texts but not able to proceed. Kindly help.

tony
  • 769
  • Actually $f$ is identically zero under those conditions, see for example https://math.stackexchange.com/questions/2216835/f0-0-with-left-fx-right-leq-left-fx-right-for-x-in-mathbbr or https://math.stackexchange.com/questions/473342/differentiable-and-continuous-functions-on-0-1-with-weird-conditions – Martin R Sep 04 '20 at 14:57

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