# Problem 3.

Point $M$ lies on the parabola $y=2 x^{2}-3 x+4$, and point $F$ lies on the line $y=3 x-4$.

Find the minimum value of $M F$.
# Solution

Let's draw a tangent to the parabola parallel to the line $\boldsymbol{y}=3 \boldsymbol{x}-4$. We will find the abscissa of the point of tangency:

$$
f^{\prime}(x)=4 x-3=3 \Rightarrow x=1.5
$$

The point $A(1.5 ; 4)$ is the point of tangency of the parabola with the line parallel to $y=3 x-4$. We will find the distance from point $A$ to the line $y=3 x-4$ using the formula:

$$
d=\frac{|4-3 \cdot 1.5+4|}{\sqrt{9+16}}=\frac{2}{5}
$$

Answer: $\frac{2}{5}$.