1. Let's understand why a function has maximums at certain points. 2. A maximum point on a function is where the function changes from increasing to decreasing. 3. Mathematically, these points are found where the first derivative $f'(x)$ equals zero and the second derivative $f''(x)$ is negative. 4. The first derivative $f'(x)$ gives the slope of the function; at maximum points, the slope is zero because the function stops increasing and starts decreasing. 5. The second derivative $f''(x)$ tells us about the concavity; if $f''(x) < 0$, the function is concave down, confirming a maximum. 6. So, the points where $f'(x) = 0$ and $f''(x) < 0$ are local maxima. 7. This explains why the function has maximums at those points.