Findind the biggest $k$ such that $e^x\ge1+kx^2$

Question

Consider the following functions: $f(x)=e^x$ and $g(x)=1+kx^2$, with $k \in \mathbb{R}^+$. Obviously, if $k=1$, $e^x \ge 1+x^2$.
My question is:
what is the biggest $k$ such that

$$e^x \ge 1+kx^2 \quad \forall x \in [0,+\infty[ $$

My attempt: $$f(x):=\frac{e^x-1}{x^2}\ge k $$ Taking the derivative, I get: $$ f'(x)=0 \iff (x-2)e^x=-2 \iff x=2+W(-2/e^2)$$

Answer 1

Consider the function $$f(x)=\frac {e^x-1}{x^2}-k$$ for which $$f'(x)=\frac{e^x (x-2)+2}{x^3}$$ cancels at $$x_*=2+W\left(-\frac{2}{e^2}\right)\implies f(x_*)=-\frac{1}{W\left(-\frac{2}{e^2}\right) \left(2+W\left(-\frac{2}{e^2}\right)\right)}-k$$ Then $k$