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 = -2, \quad b = -1, \quad c = 21

Because $a = -2 < 0$ , the parabola opens downwards ( $\cap$ ). Therefore, the function achieves an absolute maximum value at its vertex.

Step 2: Complete the Square

Factor out the leading coefficient $-2$ from the $x$ -terms:

f(x) = -2\left(x^2 + \frac{1}{2}x\right) + 21

Complete the square inside the parenthesis by adding $\left(\frac{1}{4}\right)^2 = \frac{1}{16}$ . Adding $\frac{1}{16}$ inside corresponds to adding $-2 \times \frac{1}{16} = -\frac{1}{8}$ , so we must add $\frac{1}{8}$ outside:

\begin{aligned} f(x) &= -2\left(x^2 + \frac{1}{2}x + \frac{1}{16}\right) + 21 + \frac{1}{8} \\ &= -2\left(x + \frac{1}{4}\right)^2 + \frac{168 + 1}{8} \\ &= -2\left(x + \frac{1}{4}\right)^2 + \frac{169}{8} \end{aligned}

Step 3: Determine the Extremum Point

Because $-2\left(x + \frac{1}{4}\right)^2 \le 0$ for all real numbers $x$ , the maximum is achieved when:

x + \frac{1}{4} = 0 \implies x = -\frac{1}{4} = -0.25

Step 4: Calculate the Maximum Value

Substituting $x = -\frac{1}{4}$ :

f\left(-\frac{1}{4}\right) = -2(0)^2 + \frac{169}{8} = \frac{169}{8} = 21.125