Maxima and Minima
A "peak" is a Relative Maximum and a "valley" is a Relative Minimum. We find them by checking the signs of the derivative around critical points:
- If \(f'(x)\) changes from positive to negative, you have a Maximum.
- If \(f'(x)\) changes from negative to positive, you have a Minimum.
Worked Examples
Example 1: Finding the Peak
Find the relative extrema of \(f(x) = -x^2 + 6x + 1\).
- \(f'(x) = -2x + 6\).
- Critical point: \(-2x + 6 = 0 \implies x = 3\).
- To the left (\(x=2\)): \(f'(2) = 2\) (Positive).
- To the right (\(x=4\)): \(f'(4) = -2\) (Negative).
- Result: Relative Maximum at \(x=3\).
The Bridge to Quantum Mechanics
Quantum systems always seek the state of Lowest Energy (the ground state). Finding this state is a "Minimum" problem. We vary the parameters of a trial wavefunction and use the derivative to find where the energy is minimized. This is known as the Variational Method, and it is how we calculate the energies of complex atoms and molecules where the Schrödinger Equation cannot be solved exactly.