Aritmética

APPLICATIONS OF DERIVATIVES
INCREASING AND DECREASING FUNCTIONS
Let fx be a function defined on an interval I and let x1 and x2 be any two points in I.
1. If f(x1)<f(x2) whenever x1<x2, then f(x) is said to be increasing on I
X
X
y
y
X2
X2
X
X
X11
X11
T
T

f (x2)
f (x2)
f (x1)
f (x1)
m>0
m>0

X01
X01

2. If f(x1)>f(x2) whenever x1<x2, then f(x)is said to be decreasing on I
y
y

f (x1)
f (x1)

T
T
m<0
m<0
f (x2)
f (x2)
X2
X2
X01
X01
X1
X1
X
X

If we analyze the slope of the tangent line to the curve when the graph is going up its slope is positive.  If the graph is going down, then the slope will be negative.  Since slope is the derivative of the function, we can relate increasing and decreasing functionwith its derivative.

Suppose that f(x) is continuous on a,b and differentiable on (a,b)
If f´x>0 at each point x∈a,b, then fx is increasing on (a,b).
If f´x<0 at each point x∈a,b, then fx is decreasing on (a,b).

Exercises:

Find the intervals on which the function is increasing and decreasing.
1) fx=3x2-x+8





2) fx=2x3+3x2-12x





3)fx=2x3-x2+3x-1







4) fx=2x 3-x







5) fx=x-2x+2





MAXIMUM AND MINIMUM
One of the most important uses of calculus is determining minimum and maximum values. This has its applications in manufacturing, finance, engineering, and a host of other industries.
Definition. A function f(x) has a maximum value at x=a when f(a) is more thanf(x), for all the values of x on the interval of the function. Then f(a) is a maximum value of f(x).
Definition. A function f(x) has a minimum value at x=a when f(a) is less than f(x), for all the values of x on the interval of the function. Then f(a) is a minimum value of f(x).
f(a)
f(a)
PM
PM

f(a)
f(a)
PM
PM

f(a)
f(a)
Pm
Pm

Pm
Pm
f(a)
f(a)

a
a
a
a
a
a
a
aPM
PM
A function can have several maximum and minimum values. A maximum value of a function can be less than one o more than one of its minimum values. This is why we sometimes named them as “local” maximums and minimums.


Pm
Pm

PM
PM

Pm
Pm
Pm
Pm


FIRST DERIVATIVE TEST FOR LOCAL MAXIMUMS AND MINIMUMS

Suppose that c is a critical point ( f'c=0 ) of a continuousfunction, then:

If f(x) increases on the left of c and decreases on the right of c:
f(c) is a local maximum of f(x), at x=c

If f(x) decreases on the left of c and increases on the right of c:
f(c) is a local minimum of f(x), at x=c

In summary, at a critical point c, the sign of f’(x) changes and the function has a local maximum or local minimum.

Exercises. Find thefunction’s local minimum and local maximum values.
1) fx=3x2-3x+3








2) fx=x3+7x2-5x









3) fx=x55-20x33+64x-12











4) fx=x-52x+5











5) fx=xx2-9









SECOND DERIVATIVE TEST OR CONCAVITY TEST.
The secondderivative test may also be used to determine the concavity of a function as well as a function's points of inflection.
First, all points at which are found. In each of the intervals created, is then evaluated at a single point. For the intervals where the evaluated value of the function is concave down, and for all intervals between critical points where the evaluated value of the function isconcave up. The points that separate intervals of opposing concavity are points of inflection.
Exercises: Sketch the following functions using first and second derivative test.
1) y=x3-3x+3

2) fx=1-9x-6x2-x3







3) y=3x4-8x3




















MAXIMA AND MINIMA APLICATION PROBLEMS
To solve maxima and minima application...