Theory of Intelligence • First-Principles AI & STEM Education

Step-by-Step Solution

Step 1: Analyze the Leading Coefficient

Comparing with $f(x) = ax^2 + bx + c$ :

a = 3, \quad b = 6, \quad c = -13

Because $a = 3 > 0$ , the parabola opens upwards ( $\cup$ ). Hence, the function achieves an absolute minimum value at its vertex.

Step 2: Complete the Square

Factor out the leading coefficient $3$ from the $x$ -terms:

f(x) = 3(x^2 + 2x) - 13

Inside the parenthesis, complete the square by adding $\left(\frac{2}{2}\right)^2 = 1^2 = 1$ . Since adding $1$ inside corresponds to adding $3 \times 1 = 3$ , subtract $3$ outside:

\begin{aligned} f(x) &= 3(x^2 + 2x + 1) - 13 - 3 \\ &= 3(x + 1)^2 - 16 \end{aligned}

Step 3: Determine the Extremum Point

Because $3(x + 1)^2 \ge 0$ for all real numbers $x$ , the minimum occurs when the square vanishes:

x + 1 = 0 \implies x = -1

Step 4: Calculate the Minimum Value

Substituting $x = -1$ :

f(-1) = 3(0)^2 - 16 = -16