If $$x$$ is a real number, find the smallest value of the expression $$3x^2 - 4x + 7$$.

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To find the smallest value of the quadratic expression $$3x^2 - 4x + 7$$, we use the formula for the minimum (or maximum) value of a quadratic equation. For a quadratic equation in the form $$ax^2 + bx + c$$, the minimum or maximum occurs at $$x = -\frac{b}{2a}$$.

Here, $$a = 3$$, $$b = -4$$, and $$c = 7$$. Substituting into the formula:  
$$
x = -\frac{-4}{2 \cdot 3} = \frac{4}{6} = \frac{2}{3}
$$

Now substitute $$x = \frac{2}{3}$$ into the expression to find its value:  
$$
3\left(\frac{2}{3}\right)^2 - 4\left(\frac{2}{3}\right) + 7
$$

Simplify step by step:  
$$
= 3 \cdot \frac{4}{9} - \frac{8}{3} + 7
= \frac{12}{9} - \frac{24}{9} + \frac{63}{9}
= \frac{51}{9} = \frac{17}{3}
$$

Thus, the smallest value of the expression is **$$\frac{17}{3}$$**.

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