4.2 Critical PointsFri Dec 11

Do Now

Find the derivative of each

1)

2)

Test 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

Calculator to solve equations

• 1) Graph it (y = )

• 2) 2nd -> calc -> zeros

• 3) left bound / right bound– Click to the left and right of the zero– Guess: hit enter near the zero

Using the Solver Function

• Math -> Solver

• Rewrite the equation so it is equal to 0

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

• x 2 + 6 x - 7 = 0(x + 7)(x - 1) = 0x = -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

Closure

• Find the critical numbers of

• HW: p.222 #3-17 odds

4.2 Extreme ValuesMon Dec 14

• Do Now

• Find the critical points of each function

• 1)

• 2)

HW Review: p.222 #3-17 odds

• 3) x = 1• 5) x = -3, 6• 7) x = 2• 9) x = -1, 1 • 11) t = 3, -1• 13) x = -1, 0, 1, sqrt(2/3), -sqrt(2/3)• 15) npi/2• 17) x = 1/e

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– 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 y is the max

The smallest y is the min

Ex

• Find the extrema of the function on [0,6]

Ex 2

• Find the max of the function on [-1, 2]

Ex 3

• Find the extreme values of the function on [1, 4]

Ex 4

• Find the min and max of the function on [0, 2pi]

You try

• Find the extrema of the given function on the indicated interval

• 1)

• 2)

Closure

• Find the min and max of the function on given interval

• HW: p.223 #1 21 29-41 47 51 55 odds

4.2 Rolle’s TheoremTues Dec 15

• Do Now

• Find any critical points for each function

• 1)

• 2)

HW Review: p.222 #1 21 29-41 47 51 55

• 1) a) 3 b) 6 c) max at 5 d) varies• 21) a) c = 2b) f(0)=f(4)=1

c) max: 1, min: -3 d) max: 1, min: -2• 29) min: (-1, 3), max: (2, 21)• 31) min: (0,0) max: (3, 9)• 33) min: (4, -24) max: (6, 8)• 35) min: (1, 5) max: (2, 28)• 37) min: (2, -128)max: (-2, 128)

39 41 47 51 55

• 39) min: (6, 18.5)max: (5, 26)• 41) min: (1, -1) max: (0,0) (3, 0)• 47) min: (0,0) (pi/2, 0) max: (pi/4, 1/2)• 51) min: (pi/3, -.685) max: (5pi/3, 6.968• 55) min: (1,0) max: (e, .368)

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]

Practice

• Green book worksheet p.268 #33-41

• 1) Critical points– Differentiate and solve = 0

• 2) Test endpoints and c.p. into original function

Closure

• What is a critical point? How can you tell if a critical point is a local max or min?

• HW: worksheet p.268 #33-41