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 = -1, \quad b = 8, \quad c = 13

Since $a = -1 < 0$ , the parabola opens downwards ( $\cap$ ). Consequently, the function achieves an absolute maximum value at its vertex.

Step 2: Complete the Square

Factor out $-1$ from the $x$ -terms:

\begin{aligned} f(x) &= -(x^2 - 8x) + 13 \end{aligned}

Inside the parenthesis, complete the square by adding $\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16$ . Note that adding $16$ inside multiplied by $-1$ subtracts $16$ , so we must add $16$ outside to keep equality:

\begin{aligned} f(x) &= -(x^2 - 8x + 16) + 13 + 16 \\ &= -(x - 4)^2 + 29 \end{aligned}

Step 3: Determine the Extremum Point

Because $(x - 4)^2 \ge 0$ for every real number $x$ , the negative quantity $-(x - 4)^2 \le 0$ achieves its maximum value of $0$ when:

x - 4 = 0 \implies x = 4

Step 4: Calculate the Maximum Value

Substituting $x = 4$ :

f(4) = -(4 - 4)^2 + 29 = 0 + 29 = 29