1.1 SIMPLE FUNCTIONS AND EQUATIONS
xxφ 1 (x) φ 2 (x)β 1 β 1β 2β 2Figure 1.1 Two curvesφ 1 (x)andφ 2 (x), both with zero derivatives at the
same values ofx, but with different numbers of real solutions toφi(x)=0.next chapter. The first of these is the notion of the derivative of a function, and
the second is a result known as Rolle’s theorem.
Thederivativef′(x) of a functionf(x) measures the slope of the tangent tothe graph off(x) at that value ofx(see figure 2.1 in the next chapter). For
the moment, the reader with no prior knowledge of calculus is asked to accept
that the derivative ofaxnisnaxn−^1 , so that the derivativeg′(x)ofthecurve
g(x)=4x^3 +3x^2 − 6 x−1 is given byg′(x)=12x^2 +6x−6. Similar expressions
for the derivatives of other polynomials are used later in this chapter.
Rolle’s theorem states that iff(x) has equal values at two different values ofxthen at some point between these twox-values its derivative is equal to zero; i.e.
the tangent to its graph is parallel to thex-axis at that point (see figure 2.2).
Having briefly mentioned the derivative of a function and Rolle’s theorem, wenow use them to establish whetherg(x) has one or three real zeros. Ifg(x)=0
does have three real rootsαk,i.e.g(αk)=0fork=1, 2 ,3, then it follows from
Rolle’s theorem that between any consecutive pair of them (sayα 1 andα 2 )there
must be some real value ofxat whichg′(x) = 0. Similarly, there must be a further
zero ofg′(x) lying betweenα 2 andα 3. Thus anecessarycondition for three real
roots ofg(x) = 0 is thatg′(x) = 0 itself has two real roots.
However, this condition on the number of roots ofg′(x) = 0, whilst necessary,is notsufficientto guarantee three real roots ofg(x) = 0. This can be seen by
inspecting the cubic curves in figure 1.1. For each of the two functionsφ 1 (x)and
φ 2 (x), the derivative is equal to zero at bothx=β 1 andx=β 2. Clearly, though,
φ 2 (x) = 0 has three real roots whilstφ 1 (x) = 0 has only one. It is easy to see that
the crucial difference is thatφ 1 (β 1 )andφ 1 (β 2 ) have the same sign, whilstφ 2 (β 1 )
andφ 2 (β 2 ) have opposite signs.
It will be apparent that for some equations,φ(x)=0say,φ′(x) equals zero