Differential Calculus

Maxima, minima, inflexions all have zero gradient.

$f''(x) > 0$ ⇒ minimum (concave up); $f''(x) < 0$ ⇒ maximum (concave down).

$f'(x) > 0$ rising; $f'(x) < 0$ falling.

Concavity changes sign.

1) write quantity, 2) reduce to one variable via a constraint, 3) set $f'=0$, 4) classify with $f''$.

2nd derivative: f''>0 min, f''<0 max, =0 test further (inflexion?).

f'>0 increasing; f''>0 concave up. Point of inflexion where concavity changes.

Write the quantity in ONE variable, differentiate, set =0, justify max/min, check endpoints.

f''(x)=0 does NOT guarantee inflexion — concavity must actually change sign.