The problem is as follows :
Given a continuously differentiable function $f$ on $[0,+\infty)$ such that $\begin{cases} f(0)=f'_d(0)=0 \\ \exists a \in \mathbb{R}^{+\star} \quad : \quad f(a)<0 \text{ and } f'(a)>0 \end{cases}$
it is asked to prove that the graph of $f$ has a tangent passing through the origin. (other than $y = 0$)
my work :
we should equivalently justify the existence of $c>0$ such that $f'(c)=\dfrac{f(c)}{c}$.
I defined $g(x):=\dfrac{f(x)}{x}$ for $x>0$ with $g(0)=0$.
having the idea of applying Rolle on $g$ which is continous on $[0,+\infty)$ , differentiable on $(0,+\infty)$.
the problem is Rolle between $0$ and what ?
thanks for any advice.
define g(x)=f(x)/x when x>0, and 0 when x=0. then, g is continuous in [0, INFTY), and differentiable in (0, INFTY). So, there exist a minimum value g(m) when m is in [0, a]. m is not 0(g(0)>g(a)) and m is not a (g'(a)>0). so, 0<m<a and g'(m)=0 by Fermat Thm. so, f'(m)=f(m)/m