Extremes, Tangents and Applications

Equation of the tangent

The equation of the tangent to the curve of f at the point with abscissa a is: y = f'(a) * (x - a) + f(a)

Example: f(x) = x², a = 1. We have f(1) = 1 and f'(1) = 2. Tangent: y = 2(x - 1) + 1 = 2x - 1

Local extrema

If f' is zero at a and changes sign, then f has a local extremum at a: a maximum if f' changes from positive to negative, a minimum if f' changes from negative to positive.

Example: f(x) = -x² + 4x. f'(x) = -2x + 4 = -2(x - 2) f' is positive before x = 2 and negative afterwards: f therefore has a local maximum at x = 2, with f(2) = -4 + 8 = 4.

A practical application: optimising an area

A rectangle has a fixed perimeter of 20. If x denotes one side, the other is 10 – x, and the area is: A(x) = x(10 – x) = –x² + 10x, for x in ]0; 10[ A'(x) = -2x + 10, which is zero at x = 5. A' is positive before 5 and negative afterwards: the area is maximised at x = 5 (the rectangle is then a square), with A(5) = 25.

Classic pitfall

The fact that $f'(a) = 0$ is not sufficient to guarantee a local extremum: one must verify that the sign of $f'$ does indeed change on either side of $a$. The classic counterexample is f(x) = x³ at a = 0: f'(0) = 0 but f' = 3x² remains positive (or zero) on either side, so there is no extremum at 0, only an inflection point.