The mileage of a certain car can be approximated by:

3 20.00015 0.032 1.8 1.7m v v v v

At what speed should you drive the car to obtain the best gas mileage?

Of course, this problem isn’t entirely realistic, since it is unlikely that you would have an equation like this for your car.

3 20.00015 0.032 1.8 1.7m v v v v

We could solve the problem graphically:

3 20.00015 0.032 1.8 1.7m v v v v

We could solve the problem graphically:

On the TI-84, we use 2nd CALC 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

3 20.00015 0.032 1.8 1.7m v v v v

We could solve the problem graphically:

On the TI-89, we use F5 (math), 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

The car will get approximately 32 miles per gallon when driven at 38.6 miles per hour.

3 20.00015 0.032 1.8 1.7m v v v v

Notice that at the top of the curve, the horizontal tangent has a slope of zero.

Traditionally, this fact has been used both as an aid to graphing by hand and as a method to find maximum (and minimum) values of functions.

Slope of the Tangent Line

• f ’(a) is the slope of the tangent line to the graph of f at x = a.

This tangent slope interpretation of f has several useful geometric implications.

The values of the derivative enables us to detect when a function is increasing or decreasing.

• If f ’(x) > 0 on an interval (a, b) then f is increasing on that interval.

• If f ’(x) < 0 on an interval (a, b) then f is decreasing on that interval.

A number c in the interior of the domain of a function f is called a critical number if either:

f ’(c) = 0 f ’(c) does not exist

c cPoint (c, f(c)) is called a critical point of the

graph of f.

• The only inputs, x = c, at which the derivative of a function can change sign (+ → – or – → +) are where f ‘(c) is at the critical numbers of:

00 or undefinedundefined

• A local maximum (or relative maximum) for a function f occurs at a point x = c if f(c) is the largest value of f in some interval (a < c < b) centered at x = c.

local max.

• A global maximum (or absolute maximum) for a function f occurs at a point x = c if f(c) is the largest value of f for every x in the domain.

absolute max.

• A local minimum (or relative minimum) for a function f occurs at a point x = c if f(c) is the smallest value of f in some interval (a < c < b) centered at x = c.

local min.

• A global minimum (or absolute minimum) for a function f occurs at a point x = c if f(c) is the smallest value of f for every x in the domain.

absolute min.

• A function can have at most one global max. and one global min., though this value can be assumed at many points.

• Example: f(x) = sin x has a global max. value of 1 at many inputs x.

• Maximum and minimum values of a function are collectively referred to as extreme values (or extrema).

• A critical point with zero derivative but no maximum or min. is called a plateau point.

Example 1: Find critical numbers for the given 4 2( ) 2f x x x

3'( ) 4 4f x x x 34 4 0x x 24 ( 1) 0x x

4 ( 1)( 1) 0x x x

So '( ) 0 at 1, 0 and 1f x x x x

Remember, critical numbers occur if the derivative does not exist or is

zero.

Our critical numbers!!

Example 1: 4 2( ) 2f x x x

3'( 2) 4( 2) 4( 2) 24f

3'(2) 4(2) 4(2) 24f

3'( 0.5) 4( 0.5) 4( 0.5) 1.5f 3'(0.5) 4(0.5) 4(0.5) 1.5f

A number-line graph for f and f ’ helps. It is a convenient way to sketch a

graph.Remember, if f ’<0, then f(x) is decreasing and if f ’(x)>0 f(x) is

increasing.

x –1 10

f '(x) – + – +

Example 1: 4 2( ) 2f x x x Recall, local min. at x=c if f(c) is the

smallest value in the interval.

x –1 10

f '(x) – + – +

Recall, local max. at x=c if f(c) is the largest value in the interval.

min. min.

max.f (x)

Extreme values can be in the interior or the end points of a function.

2y x

,D Absolute Minimum

No AbsoluteMaximum

2y x

0,2D Absolute Minimum

AbsoluteMaximum

2y x

0,2D No Minimum

AbsoluteMaximum

2y x

0,2D No Minimum

NoMaximum

Extreme Value Theorem:

If f is continuous over a closed interval, then f has a maximum and minimum value over that interval.

Maximum & minimumat interior points

Maximum & minimumat endpoints

Maximum at interior point, minimum at endpoint

Local Extreme Values:

A local maximum is the maximum value within some open interval.

A local minimum is the minimum value within some open interval.

Absolute minimum(also local minimum)

Local maximum

Local minimum

Absolute maximum

(also local maximum)

Local minimum

Local extremes are also called relative extremes.

Local maximum

Local minimum

Notice that local extremes in the interior of the function

occur where is zero or is undefined.f f

Absolute maximum

(also local maximum)

EXAMPLE FINDING ABSOLUTE EXTREMA

Find the absolute maximum and minimum values of on the interval . 2/3f x x 2,3

2/3f x x

1

32

3f x x

3

2

3f x

x

There are no values of x that will makethe first derivative equal to zero.

The first derivative is undefined at x=0,so (0,0) is a critical point.

Because the function is defined over aclosed interval, we also must check theendpoints.

0 0f To determine if this critical point isactually a maximum or minimum, wetry points on either side, withoutpassing other critical points.

2/3f x x

1 1f 1 1f

Since 0<1, this must be at least a local minimum, and possibly a global minimum.

2,3D

At: 0x

At: 2x 2

32 2 1.5874f

At: 3x 2

33 3 2.08008f

0 0f To determine if this critical point isactually a maximum or minimum, wetry points on either side, withoutpassing other critical points.

2/3f x x

1 1f 1 1f

Since 0<1, this must be at least a local minimum, and possibly a global minimum.

2,3D

At: 0x

At: 2x 2

32 2 1.5874f

At: 3x

Absoluteminimum:

Absolutemaximum:

0,0

3,2.08

2

33 3 2.08008f

Absolute minimum (0,0)

Absolute maximum (3,2.08)

2/3f x x

Finding Maximums and Minimums Analytically:

1 Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points.

2 Find the value of the function at each critical point.

3 Find values or slopes for points between the critical points to determine if the critical points are maximums or minimums.

4 For closed intervals, check the end points as well.

Critical points are not always extremes!

3y x

0f (not an extreme)

1/3y x

is undefined.f

(not an extreme)

Find the critical numbers of each function

2

2 2

( ) ( 3)

( ) ( 4)

( ) 4

f x x x

g x x x

h x x x

Locate the absolute extrema of each function on the closed interval.

( ) 2(3 ) 1,2f x x

Locate the absolute extrema of each function on the closed interval.

2 5( ) 0,5

3

xf x

Locate the absolute extrema of each function on the closed interval.

2( ) 3 0,3f x x x

Locate the absolute extrema of each function on the closed interval.

2( ) 2 4 1,1f x x x

Locate the absolute extrema of each function on the closed interval.

3 2( ) 3 1,3f x x x

Locate the absolute extrema of each function on the closed interval.

2/3( ) 3 2 1,1f x x x

Locate the absolute extrema of each function on the closed interval.

2

2( ) 1,1

3

xf x

x

If f (x) is continuous over [a,b] and differentiable

over (a,b), then at some point c between a and b:

f b f af c

b a

Mean Value Theorem for Derivatives

The Mean Value Theorem only applies over a closed interval.

The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.

y

x0

A

B

a b

Slope of chord:

f b f a

b a

Slope of tangent:

f c

y f x

Tangent parallel to chord.

c

Find all values of c in the interval (a,b) such that

f b f af c

b a

2( ) 2,1f x x

Find all values of c in the interval (a,b) such that

f b f af c

b a

2( ) 2 1,1f x x x x

Find all values of c in the interval (a,b) such that

f b f af c

b a

3( ) 0,1f x x