P2.7 - Differentiation

The derivative $f'(x)$ is the gradient of a curve. Its sign tells whether the function rises or falls, while a stationary point is a point where $f'(x)=0$.

When $f'(c)=0$, the second derivative gives a quick local test: $f''(c)<0$ indicates a maximum and $f''(c)>0$ a minimum. If $f''(c)=0$, this test is inconclusive, so use the sign of $f'$.

For $f(x)=x^3-3x^2-9x+5$, $$f'(x)=3(x+1)(x-3).$$ Thus the stationary points are $(-1,10)$ and $(3,-22)$. The derivative is positive for $x<-1$, negative for $-1<x<3$, and positive for $x>3$; therefore the first point is a local maximum and the second a local minimum.

For a curve sketch, combine stationary coordinates and increasing/decreasing intervals with intercepts and end behaviour. In a practical optimisation on a restricted domain, compare the objective value at every feasible stationary point and included endpoint, then state the maximum or minimum with its units and context.