Final answer:
Maxima and minima can be found without derivatives by using physical principles like conservation of mechanical energy and understanding of equilibrium. By assessing conditions at different points and understanding the behavior of energy, one can avoid calculus in determining extremum points in physics problems.
Step-by-step explanation:
Finding maxima and minima without derivatives can be approached by using physical principles such as the conservation of energy, especially in physics contexts where velocity and acceleration are considered. For example, when a car goes up a slope, the conservation of mechanical energy can be used to determine the speed at the top of the slope by analyzing conditions just before the car is released and just after it reaches the top of the slope. This method avoids the complexities of calculus but requires a good understanding of the physical situation and the energies involved.
In cases where a graph or curve is involved, you might estimate the shape of the graph over smaller intervals to find relative maxima or minima, and use the tangent line to calculate instantaneous acceleration. Importantly, the second derivative test, which usually involves calculus, can sometimes be deciphered from physical insights about stability, such as determining if an equilibrium is stable or unstable without actually taking the derivative. Analyzing whether a term in an equation is positive or negative at certain critical points can indicate the nature of the extremum - maximum or minimum.