Second Derivatives

For certain functions, it is possible to differentiate twice and find the derivative of the derivative. It is often denoted as f or \frac{d^2y}{dx^2}. For example, given that f(x)=x^7-x^5 then the derivative is f and the second derivative is given by f.

How to classify stationary points:

The second derivative can tell us something about the nature of a stationary point. Suppose that we have found the x-coordinates of all of the stationary points by solving f(x)=0. For a minimum, the gradient changes from negative to 0 to positive, i.e. the gradient is increasing. If the gradient is increasing then the gradient of the gradient is positive, i.e. f. For a maximum, the gradient changes from positive to 0 to negative, i.e. the gradient is decreasing. If the gradient is decreasing then the gradient of the gradient is negative, i.e. f. By putting the x-coordinates of the stationary points into f, we can classify whether they are minima or maxima by determining whether the second derivative is positive or negative at those x-coordinates.

ExampleFind and classify the stationary points of f(x)=x^3-2x^2+x-5.

We locate the stationary points by solving f. f'(x) is given by


We can solve f'(x)=0 by factorising:

which gives x=1/3 or x=1. The corresponding y coordinates are \left(\frac{1}{3}\right)^3-2\left(\frac{1}{3}\right)^2+\frac{1}{3}-5=-\frac{131}{27} (don’t be afraid of strange fractions) and (1)^3-2(1)^2+1-5=-5. Hence, the stationary points are at (1/3,-131/27) and (1,-5). We can classify the stationary points by substituting the x coordinate of the stationary point into the second derivative and seeing if it is positive or negative. Differentiating a second time gives
f. It follows that f which is less than 0, and hence (1/3,-131/27) is a MAXIMUM. f and (1,-5) is a MINIMUM.