## The Second Derivative

The second differentiail criterion al

lows us to classify turning points of a function
$f(x)$
(points for which
$\frac{df}{dx}=0$
).
Suppose there is a function
$f(x)$
and that at some point
$x_0$
,
$\frac{df}{dx}=0$
.
Then
$(x_0, f(x_0))$
is a turning point of the function
$f(x)$
and
If
$\frac{d^2f}{dx^2} < 0$
then
$(x_0, f(x_0))$
is at a maximum for the function
$f(x)$
.
If
$\frac{d^2f}{dx^2} > 0$
then
$(x_0, f(x_0))$
is at a minimum for the function
$f(x)$
.
Example: Let
$f(x)=x^3-4x^2-3x$
. Then
$\frac{df}{dx}=3x^2-8x-3$
. Ti find the turning point set
$\frac{df}{dx}=3x^2-8x-3=(3x+1)(x-3)=0$
.
Then the turning points are
$x_1 =- \frac{1}{3}, \: x_2 =3$
.
$\frac{d^2f}{dx^2}=6x-8$
.
When
$x=- \frac{1}{3}, \: \frac{d^2f}{dx^2}=6 \times - \frac{1}{3}-8=-10 \lt 0$
.
Hence
$x_1= - \frac{1}{3}$
is at a maximum.
When
$x=3, \: \frac{d^2f}{dx^2}=6 \times 3-8=10 \gt 0$
.
Hence
$x_1= - \frac{1}{3}$
is at a minimum.

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