Exercise 1 (5 points) The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$. Consider the function $f$ defined on the interval $]0; 1]$ by $$f(x) = x(1 - \ln x)^2.$$ a. Determine an expression for the derivative of $f$ and verify that for all $x \in ]0; 1]$, $f'(x) = (\ln x + 1)(\ln x - 1)$. b. Study the variations of the function $f$ and draw its variation table on the interval $]0; 1]$ (it will be admitted that the limit of the function $f$ at 0 is zero).
Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
Consider the function $f$ defined on the interval $]0; 1]$ by
$$f(x) = x(1 - \ln x)^2.$$
a. Determine an expression for the derivative of $f$ and verify that for all $x \in ]0; 1]$, $f'(x) = (\ln x + 1)(\ln x - 1)$.
b. Study the variations of the function $f$ and draw its variation table on the interval $]0; 1]$ (it will be admitted that the limit of the function $f$ at 0 is zero).