Testing for Change
We can use the sign of the first derivative to determine if a function is going up or down:
- If \(f'(x) > 0\), the function is increasing.
- If \(f'(x) < 0\), the function is decreasing.
- If \(f'(x) = 0\), the function is at a critical point (flat).
Worked Examples
Example 1: Analyzing a Parabola
Determine where \(f(x) = x^2 - 4x + 3\) is increasing or decreasing.
- \(f'(x) = 2x - 4\).
- Set \(f'(x) = 0 \implies 2x = 4 \implies x = 2\).
- Test \(x < 2\): \(f'(1) = 2(1) - 4 = -2\) (Decreasing).
- Test \(x > 2\): \(f'(3) = 2(3) - 4 = 2\) (Increasing).
- Result: Decreasing on \((-\infty, 2)\), Increasing on \((2, \infty)\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often look at the Probability Density \(P(x) = |\psi(x)|^2\). We want to know where the particle is most likely to be found. By looking at where \(P(x)\) is increasing or decreasing, we can find the "peaks" of probability. These peaks correspond to the most probable positions of an electron in an atom.