1. **Problem:** What number exceeds its square by the maximum amount? 2. **Step 1: Define the function.** Let the number be $x$. The amount by which the number exceeds its square is given by: $$f(x) = x - x^2$$ 3. **Step 2: Find the critical points.** To find the maximum, differentiate $f(x)$ with respect to $x$: $$f'(x) = 1 - 2x$$ Set the derivative equal to zero to find critical points: $$1 - 2x = 0 \implies x = \frac{1}{2}$$ 4. **Step 3: Determine the nature of the critical point.** Find the second derivative: $$f''(x) = -2$$ Since $f''(x) = -2 < 0$, the function has a maximum at $x = \frac{1}{2}$. 5. **Step 4: Calculate the maximum amount.** Substitute $x = \frac{1}{2}$ into $f(x)$: $$f\left(\frac{1}{2}\right) = \frac{1}{2} - \left(\frac{1}{2}\right)^2 = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$$ **Answer:** The number $\frac{1}{2}$ exceeds its square by the maximum amount of $\frac{1}{4}$.