EDB — 1TN

Exercises

1. [1TN]Note:Exercise 2, written exam 4 April 2009.

   • Verify that for every $t>1$ the equation

     $$\sin x=x^t$$

     admits one and only one solution $x>0$.

   • Call $f(t)$ this solution, determine the image of the function $t$ and show that it is strictly increasing and continuous on $(1,+\infty )$.

   • Prove that $f$ is extended by continuity to $t=1$ and discuss the existence of the right derivative of the prolonged function at that point.

   Solution 1

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