Ch. 5 – Applications of Derivatives

5.1 – Extreme Values of Functions

• Absolute Extreme Values: Let f be a function with domain D. Then f(c) is the...– ...absolute maximum value on D if f(x) ≤ f(c) for

all x in D.– ...absolute minimum value on D if f(x) ≥ f(c) for

all x in D.– The absolute max and min values can be endpoints or interior

points of a domain.

• Ex: Use your grapher to find the x-coordinates of the absolute extrema for the following functions:

– Absolute min at x = -2– Absolute max at x = 1

– Absolute min at x = π– No absolute max because x = 0 and x = 2π aren’t in the

domain

2( ) 4 5, 4 1f x x x x

( ) cos , 0 2f x x x

• Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f has both a maximum and a minimum on the interval.

• Local (or Relative) Extrema: – A local maximum occurs at x = c when f(c) is larger

than the f(x) values to its left and right. – A local minimum occurs at x = c when f(c) is

smaller than the f(x) values to its left and right.

• To find the local extrema algebraically, find all points c such that f’(c) = 0 or f’ does not exist and find the values of the endpoints.

• A critical point is a point in the interior of the domain of a function f at which f’=0 or f’ does not exist.– A stationary point is a critical point where f’=0.

• Ex: Find the absolute extrema of f(x)=3x2 – 12x + 5 by hand over the interval [0, 3]. – First, find the endpoints:

• Endpoints: (0, 5) and (3, -4)– Second, find the critical points:

• f‘ = 6x – 12 = 0 at x = 2...• ...and f’(x) is defined for all x over [0, 3], so the only critical

value is at (2, -7)– With all of these points, find your absolute extrema!

• Absolute min: (2, -7)• Absolute max: (0, 5)

• Use a slope sign chart to find the x-coordinates of the local extrema for f(x) = x3 - 4x2 - 3x +2.– No endpoints here, so check critical points...

– Now make a number line with the 2 critical points. This number line represents the domain of f’.

– We’ve divided the domain of x into 3 sections; choose an x-value from each section and plug them into f’ to determine whether or not the slope of f is positive or negative.

• If the value is positive (or negative), mark a plus (or a minus) in that section of the number line

– Since the sign chart represents the slopes of f... • ...we know we have a local maximum at x = -1/3 because the slope goes

from positive to negative• ...we know we have a local minimum at x = 3 because the slope goes from

negative to positive

2'( ) 3 8 3 0f x x x (3 1)( 3) 0x x 1

3,3

x

1

3

3

'(0) 3f '( 1) 8f '(5) 32f

+ +-

• Ex: Use a grapher to find the extrema over the interval [-1, 2] for f(x) = x4 – x3 – x2 + x + 1.– Use zstandard, then change the domain to [-5, 5]– We have several extrema here. Use your calc menu

to find any local maxima and minima in the interval [-1, 2]• Endpoints: (-1, 1) and (2, 7) THESE ARE LOCAL EXTREMA!• Local maxima: (.390, 1.201), (-1, 1), and (2, 7)• Local minima: (-.640, .380) and (1, 1)

– Absolute minimum: (-.640, .380) LOWEST Y-VALUE!– Absolute maximum: (2, 7) HIGHEST Y-VALUE!– If we graph f’, we see that the local extrema

(excluding endpoints) of f are the same as the zeros of f’