10.8 Maxima and minima of functions

Most people’s lives have ups and downs that are relatively gradual, a sinuous curve with first derivatives at every point. They’re the ones who never get struck by lightning. No real idea of cataclysm at all. But the ones who do get hit experience a singular point, a discontinuity in the curve of life—do you know what the time rate of change is at a cusp? Infinity, that’s what! A-and right across the point, it’s minus infinity! How’s that for sudden change, eh? Infinite miles per hour changing to the same speed in reverse, all in… the Δ⁢t\Delta t across the point. That’s getting hit by lightning, folks.

—Thomas Pynchon, Gravity’s Rainbow. Usually somewhere around p. 660 (depending on edition). Note: some profanity has been removed.

Consider this curve,

−10-10−5-50551010−150-150−100-100−50-5005050xxf⁢(x)f(x)−x2+6⁢x+10-x^{2}+6x+10

What is the gradient at the turning point? Well in the instant where the curve’s gradient is turning from having a positive gradient to a negative one, there will be a point where the gradient is zero. Before this point, the gradient is increasing. After this point, the gradient is negative 1111 11 Note that this is only true for some curves - if the curve was the other way up, then the gradient would be negative before the point where the gradient is zero, and positive after that point.. It’s the point in the middle where there’s no change, and where the "turning point" of the curve is.

We can therefore write that for a function f⁢(x)f(x), the turning points of the curve are some of the points where (turning points are points where the sign of the derivative either side of the stationary point - i.e. when the derivative is zero - changes, so from positive to negative or negative to positive; more on how to spot the difference is given further down)

d⁢f⁢(x)x=0\frac{df(x)}{x}=0 (10.49)

How do the turning points relate to the minima and maxima of the curve? A turning point (which we can write in co-ordinate form as (x,f⁢(x))(x,f(x))) has a value of f⁢(x)f(x) that is either smaller or larger than the points around it. This means that all turning points are either local minima or maxima. What this means is that they’re bigger/smaller than the other points immediately around them, but not necessarily bigger/smaller than all the points on the whole curve.

Sometimes, for example, the curve doesn’t have a maximum or minimum because it keeps growing! This curve, for example, rides off towards the sunset of infinity:

−10-10−5-50551010−1,000-1{,}000−500-5000500500xxf⁢(x)f(x)x3−4⁢x2+6⁢x+10x^{3}-4x^{2}+6x+10

Other times, we do have a global maxima/minima 1212 12 a global maxima is a point on the curve such that there is no other point that is bigger than it , however, not all local maxima are global maxima!

−4-4−3-3−2-2−1-101122022446688xxf⁢(x)f(x)−x4−2⁢x3+2⁢x2-x^{4}-2x^{3}+2x^{2}

Another thing to note is that a local maxima requires

d2⁢f⁢(x)d⁢x2<0⁢ and ⁢d⁢yd⁢x=0\frac{d^{2}f(x)}{dx^{2}}<0\textit{ and }\frac{dy}{dx}=0 (10.50)

and a local minima requires that

d2⁢f⁢(x)d⁢x2>0⁢ and ⁢d⁢yd⁢x=0\frac{d^{2}f(x)}{dx^{2}}>0\textit{ and }\frac{dy}{dx}=0 (10.51)

If we have instead that

d2⁢f⁢(x)d⁢x2=0⁢ and ⁢d⁢yd⁢x=0\frac{d^{2}f(x)}{dx^{2}}=0\textit{ and }\frac{dy}{dx}=0 (10.52)

we don’t have a turning point - we have a stationary point!