What the app does
f′ is computed symbolically — exactly, by the product, quotient and chain rules, not by approximation. Its sign gives the direction of travel: positive rising, negative falling. Turning points are found three ways, because looking for zeros of f′ alone misses two of them: where f′ crosses zero, where it jumps sign without ever being zero (the corner of |x|), and at closed endpoints of the domain, where f′ need not vanish at all. Everything the function reaches is then weighed against the values it only approaches, to decide the absolute maximum and minimum.
Why it matters
This is where the interesting points live. Note the distinction the last step draws: a value that is reached is a maximum, while one the curve only closes in on — like the 1 that sin(x)/x approaches at its hole — is a supremum, and saying so is not pedantry but the difference between an attainable answer and an unattainable one.
Read further
- Derivative Wikipedia
- Maximum and minimum Wikipedia
- Fermat's theorem (stationary points) Wikipedia
- Minimum and maximum values Paul's Online Notes