Application to finding extremal points.

The following proposition is often taught without proof. A nice proof is writtenusing the Taylor developments of the function $f$ at $x_0$ (cf Chapter 10 section Taylor series).

Proposition 6.10.11   Let $f$ be a function defined on a neighborhood of $x_0$ . Suppose that $f$ is differentiable (at least) twice at $x_0$ and that $f'(x_0) =0$ .

1. If $f''(x_0)<0$ , then $f$ has a maximum at $x_0$ .
2. If $f''(x_0)>0$ , then $f$ has a minimum at $x_0$ .

If $f''(x_0)=0$ , then the behavior of the function $f$ at $x_0$ depends on the derivatives of higher order.

Proposition 6.10.12   Let $f$ be a function defined on a neighborhood of $x_0$ . Suppose that $f$ has enough derivatives on this neighborhood and that $f'(x_0)=f''(x_0)=0$ . Denote by $n$ the order of the first derivative which does not vanish at $x_0$ .

i.

If $n$ even and $f^{(n)}(x_0)<0$ , then $f$ has an maximum at $x_0$ .

ii.

If $n$ even and $f^{(n)}(x_0)>0$ , then $f$ has an minimum at $x_0$ .

iii.

If $n$ odd, then $f$ has a point of inflexion at $x_0$ .

Example 6.10.13       

1. Let $f(x)=x^4$ . Then $f'(x)=4x^3$ , $f''(x)=12x^2$ . Both derivatives vanish at 0. The first occurrence of a derivative which does not vanish at 0 is the $4^{th}$ derivative and $f^{(4)}(0)=24$ . It follows that $f$ has a minimum at 0 (see Figure 13(a)).
2. Let $f(x)=x^5$ . Then $f'(x)=5x^4$ , $f''(x)=20x^3$ . Both derivatives vanish at 0. The first occurrence of a derivative which does not vanish at 0 is the $5^{th}$ derivative. This order is odd, hence $f$ has a point of inflexion at 0 (see Figure 13(b)).

Figure 13:

Example 6.10.14   Let $f(x)=\sin x + \sin 2x$ ; the graph of $f$ is displayed on Figure 14.

The function $f$ is defined by a trigonometric polynomial, hence it is differentiable over $\mathbb{R}$ . We have: $f'(x)= \cos x + 2 \cos 2x$ . Now check that this first derivative vanishes at $x_0 = -0.935929 \;$   rd .

Figure 14: Extrema of a trigonometric function.

We have: $f''(x)= -\sin x - 4 \sin 2x$ ; then $f''(x_0)>0$ . Therefore the function $f$ has a minimum at $x_0$ .Similarly, we see that $f'$ vanishes at $x_1=0.935929 \;$   rd ; there we have $f''(x_0)<0$ , i.e. $f$ has a maximum at $x_1$ .

Of course, the above computations are not sufficient for deciding whether these extrema are absolute or local.