section 5.1 – increasing and decreasing functions the first derivative test (max/min) and its...
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Section 5.1 – Increasing and Decreasing Functions
The First Derivative Test (Max/Min)and its documentation
5.2
The Theory First……
THE FIRST DERIVATIVE TEST
If c is a critical number and f ‘ changes signs at x = c, then
• f has a local minimum at x = c if f ‘ changes from neg to pos.
• f has a local maximum at x = c if f ‘ changes from pos to neg
2Let f be a function given by f x 2ln x 3 x with domain
3 x 5. Find the x-coordinate of each relative maximum point
and each relative minimum point of f. Justify your answer.
2
2xf ' x 2 1
x 3
2
2 2
2x x 30 2
x 3 x 3
2
2
x 4x 30
x 3
2
x 3 x 10
x 3
x 3,1
_+
_
There is a rel min at x = 1 because f ‘ changes from neg to pos
There is a rel max at x = 3 because f ‘ changes from pos to neg
1 3-3 5
f ' x
NO CALCULATOR
The Theory…Part II
EXTREME VALUE THEOREMIf a function f is continuous on a closed interval [a, b] then fhas a global (absolute) maximum and a global (absolute) minimum value on [a, b].
GLOBAL (ABSOLUTE) EXTREMA
A function f has:
•A global maximum value f(c) at x = c if f(x) < f(c) for every x in the domain of f.
•A global minimum value f(c) at x = c if f(x) > f(c) for every x in the domain of f.
The Realities…..
On [1, 8], the graph of any continuous function HAS to•Have an abs max•Have an abs min
2xLet f be a function given by f x 2xe . Find the absolute
minimum value of f. Justify your answer is an absolute
minimum.
2x 2xf ' x 2 e 2e 2x
2x0 2 e 1 2x 1
x2
+_
There is an abs min at x = -1/2
1
2
f ' x
1
221 1
2 2f 2 e
-1
The minimum value is e
1 1f ' x 0 on , and f ' x 0 on ,
2 2
2 3If the derivative of the function f is f ' x 3 x 2 x 1 x 3
then find the value(s) of x at which there is a local minimum.
2 30 3 x 2 x 1 x 3 x 2, 1, 3
+__+
Justify your answer.
A local min occurs at x = 3 since f ' x
changes from neg to pos
-2 -1 3
f ' x
2
A particle moves on the x-axis in such a way that its position
at time t, t > 0, is given by x t ln t . At what value of t
does the velocity of the particle attain its maximum. Justify
your answer.
2ln tx ' t v t
t
2
2t 1 2ln t
tv ' t
t
2
2 1 ln t0
t
ln t 1
t e
+ _
The max occurs at t = e since
f ' x 0 on 0, e
and f ' x 0 on e,
e0
f ' x
2lnx
Find a relative maximum of f xx
Justify your answer.
2
2
12 lnx x 1 lnx
xf ' x
x
2
2
2 lnx lnxf ' x
x
2
lnx 2 lnx0
x
lnx 0
x 1
2
lnx 2
x e
_ + _
2
2
A rel max occurs at e since
f ' changes from pos to neg at e
2 22
22 2 2
2
lne 2lne 4f e
e e e4
The maximum value is e
1 2e
f ' x
Find the absolute minimum of f x x lnx. Justify your answer.
1f ' x 1 lnx x
x
0 lnx 1
1 lnx 1
xe
+_
1An abs min occurs at
e1
since f ' x 0 on ,e
1and f ' x 0 on ,
e
1 1 1 1The minimum value is f ln
e e e e
1
e
f ' x
GRAPHING CALCULATOR REQUIRED
2Let f be the function defined by f x ln x 1 sin x
for 0 x 3.
a Find the x-intercepts of the graph of f
b Find the intervals on which f is increasing
c Find the absolute maximum and the absolute minimum
value
Round all of your answer
of f. Justify your answ
s to three decimal pl
er.
aces.
2Let f be the function defined by f x ln x 1 sin x
for 0 x 3.
a Find the x-intercepts of the graph of f
x = 1.684x = 0.964
x = 0
2Let f be the function defined by f x ln x 1 sin x
for 0 x 3.
b Find the intervals on which f is increasing
[0, 0.398), (1.351, 3]
2Let f be the function defined by f x ln x 1 sin x
for 0 x 3.
c Find the absolute maximum and the absolute minimum
value of f. Justify your answer.
From part b, f ' x 0 when x 0.398,1.351
0 0
0.398 0.185
1.351 0.098
endpoint
end
f
f f ' x 0
f f ' x 0
f 3 p1.36 o nt6 i
The absolute max is 1.366 and occurs when x = 3The absolute min is –0.098 and occurs when x = 1.351
kx
k
Find the coordinates of the absolute maximum point for
the curve y xe where k is a fixed positive number.
1 1 1 e 1 1A. , B. , C. , D. 0, 0 E. DNE
k ke k k k e
Let k = 2 and proceed2xy xe
2x 2xdy1 e 2e x
dx
2x 2x
1 2x0
e e
1x
2
12
21 1y e
2 2e
3
A particle starts at time t = 0 and moves along a number
line so that its position, at time t 0, is given by
x t t 2 t 6 . The particle is moving left for:
A. t 3 B. 2 t 6 C. 3 t 6 D. 0 t 3 E. t 6
3 2x' t 1 t 6 3 t 6 1 t 2
2x' t t 6 t 6 3 t 2
2x' t t 6 4t 12
t 3, 6
3 6
_ + +
2 2xIf f x x e , then the graph of f is increasing for all x such that
1A. 0 x 1 B. 0 x C. 0 x 2 D. x 0 E. x 0
2
2x 2x 2f ' x 2x e 2e x
2x 2f ' x e 2x 2x 2xf ' x 2 e x 1 x
x 0,1
_ + _
0 1
f ' x
The sale of lumber S (in millions of square feet) for the years
1980 to 1990 is modeled by the function
S t 0.46cos 0.45t 3.15 3.4 where t is the time in years
with t = 0 corresponding to the beginning of
1980. Determine the
year when lumber sales were increasing at the greatest rate
A. 1982 B. 1983 C. 1984 D. 1985 E. 1986
CALCULATOR REQUIRED
t = 3.472
2 cFor what value of x will x have a relative minimum at x 1?
xA. 4 B. 2 C. 2 D. 4 E. none of these
22
c cf x x f ' x 2x
x x
2
c0 2x
x
2
1
c2 10