How do I Determine a Curve Maxima?
- 1). Compute the general first derivative. For example, if the curve is described by the function (x^4)/4 -- (2x^3)/3 --(3x^2)/2, the first derivative equals x^3 -2x^2 -3x.
- 2). Set the first derivative equal to zero and solve. This is usually best accomplished by factoring. Continuing the example, the factored form of the first derivative is x * (x-3) * (x+1). The first derivative therefore is equal to zero when x = 0,3,-1. The points on the curve that correspond to these x-values are local minima, local maxima or saddle points.
- 3). Compute the general second derivative. Continuing the example, the second derivative is 3x^2 - 4x -- 3.
- 4). Plug in the x-values where the first derivative equals zero to the second derivative function. The sign of the second derivative at these x-values tells you whether they are local minima (second derivative positive), local maxima (second derivative negative), or saddles (second derivative zero). In the example above, when x = 0 the second derivative equals -3; when x = 3 the second derivative equals 12; when x = -1 the second derivative equals 4. The curve therefore has a local maxima at x = 0 and local minima at x = -3,-1.
