Example 2. Given that $\mathrm{AD}$ is the median of $\triangle \mathrm{ABC}$, $\mathrm{E}$ is a point on $AB$, and $CE$ intersects $AD$ at $F$.
Find the maximum area of $\triangle EDF$.

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Slightly solve: Taking D as the origin and BC as the X-axis to establish a coordinate system, $\frac{\mathrm{AE}}{\mathrm{EB}}=\lambda$, write out the coordinates of $\mathrm{E}$ and $\mathrm{F}$, and get the area of $\triangle \mathrm{EOF}$
$$
S(\lambda)=-\frac{b c \lambda}{2(1+\lambda)(1+2 \lambda)} .
$$

Using the discriminant or differential method, it is known that when $\lambda=\frac{\sqrt{2}}{2}$