Transcript 4.2 Extreme Values Mon Dec 10

4.2 Critical Points
Mon Oct 19
Do Now
Find the derivative of each
1)
f (x) = x - 3x + 2
2)
f (x) = sin x cos x
4
3
HW Review
Critical points
• A number a in the domain of a given
function f(x) is called a critical point of
f(x) if f '(a) = 0 or f ’(x) is undefined at x
= a.
• To find a critical point, we find the 1st
derivative and set it equal to 0
• Example 1: Find the critical point(s) of
the polynomial function f given by f(x)
= x 3 - 3x + 5
Solution
• Solution to Example 1.
– The first derivative f ' is given by
f '(x) = 3 x 2 - 3
– f '(x) is defined for all real numbers. Let us
now solve f '(x) = 0
• 3 x 2 - 3 = 0 = 3(x-1)(x+1) =0
• x = 1 or x = -1
– Since x = 1 and x = -1 are in the domain of
f they are both critical points.
• Example 2: Find the critical point(s) of the
rational function f defined by
f(x) = (x 2 + 7 ) / (x + 3)
• Solution to Example 2.
– Note that the domain of f is the set of all real
numbers except -3.
– The first derivative of f is given by
f '(x) = [ 2x (x + 3) - (x 2 + 7 )(1) ] / (x + 3) 2
– Simplify to obtain f '(x) = [ x 2 + 6 x - 7 ] / (x + 3) 2
– Solving f '(x) = 0
•
x2 + 6 x - 7 = 0
(x + 7)(x - 1) = 0
x = -7 or x = 1
• f '(x) is undefined at x = -3 however x = -3 is not included
in the domain of f and cannot be a critical point.
• The only criticalpoints of f are x = -7 and x = 1.
• Example 3: Find the critical point(s) of
function f defined by
f(x) = (x - 2) 2/3 + 3
• Solution to Example 3.
– Note that the domain of f is the set of all
real numbers.
– f '(x) = (2/3)(x - 2) -1/3
= 2 / [ 3(x - 2) 1/3]
– f ’(x) is undefined at x = 2 and since x = 2
is in the domain of f it is a critical point.
You try: Find the critical points
•
•
•
•
a) f(x) = 2x 3 - 6 x - 13
b) f(x) = (x - 3) 3 - 5
c) f(x) = x 1/3 + 2
d) f(x) = x / (x + 4)
answers
•
•
•
•
A) 1, -1
B) 3
C) 0
D) none
Extreme Values
• Extreme values refer to the minimum or
maximum value of a function
• There are two types of extreme values:
– Absolute extrema: the min or max value of
the entire function or interval
– Local extrema: the min or max value of a
piece of a function
Absolute vs Local (pics)
• Absolute extrema may or may not exist
• Local extrema always exist
How to find absolute extrema
• 1) Find all critical points in an interval.
• 2) Test all critical points and endpoints
into the original function
• 3) The biggest is the absolute max
The smallest is the absolute min
Ex
• Find the extrema of the function on [0,6]
f (x) = 2x -15x + 24 x + 7
3
2
Ex 2
• Find the max of the function on [-1, 2]
f (x) =1 - (x -1)
2/3
Ex 3
• Find the extreme values of the function
2
on [1, 4]
f (x) = x - 8ln x
Ex 4
• Find the min and max of the function on
2
[0, 2pi]
f (x) = sin x + cos x
You try
• Find the extrema of the given function on the
indicated interval
• 1) f (x) = x 4 - 8x 2 + 2, [-1,3]
• 2)
f (x) = x , [-1,3]
2/3
Rolle’s Theorem
• Assume that f(x) is continuous on [a,b] and
differentiable on (a,b). If f(a) = f(b), then there
exists a number c between a and b such that
f’(c) = 0
Ex
• Verify Rolle’s Theorem on [-2, 2]
f (x) = x - x
4
2
Closure
• Find the min and max of the function on
given interval
f (x) = 2x + 4x + 5, [0,2]
2
• HW: p.222 #1 5 7 21 35 47 55 65 78 83