Maximum and Minimum Absolute Maximum or Minimum A
Maximum and Minimum
Absolute Maximum or Minimum �A function f has an absolute maximum at c if f(c)≥f(x) for all x in the domain. The number f(c) is called the maximum value of f �A function f has an absolute minimum at c if f(c)≤f(x) for all x in the domain. The number f(c) is called the minimum value of f �These absolute maximum or minimum values are called the extreme values of f
Local (Relative) Maximum or Minimum �Maximum or minimum in a smaller area �Occur at any “hill” or “valley” on a function
Examples: Tell all max and min values for each 1) f(x) = cos x 2) f(x) = x 2 3) f(x) = x 3 4) f(x) = 3 x 4 – 16 x 3+ 18 x 2 on interval [-1, 4]
![Extreme Value Theorem �If f is continuous on a closed interval [ a, b],](http://slidetodoc.com/presentation_image_h/f64ad031f3f07d9bd49eb526ae161124/image-5.jpg "Extreme Value Theorem �If f is continuous on a closed interval [ a, b],")
Extreme Value Theorem �If f is continuous on a closed interval [ a, b], then f attains an absolute maximum f(c) and an absolute minimum f(d) at some number c and d in [a, b] - more than one extreme may exist - if f is not continuous, may not have an extreme - if the interval is not closed, there may not be an extreme
Fermat’s Theorm �If f has a local maximum or minimum at c, and if f’(c) exists, then f’(c) = 0 �Be careful, at every local max or min the tangent is horizontal, but not every horizontal tangent is a local max or min �Also may be a max or min at locations where f’ does not exist
Critical Number �A number where c in the domain of f such that either f’(c) = 0 or f’(c) does not exist
Example �Find the critical numbers of
Finding Absolute Max and Min on a Closed Interval Find the critical numbers 2) Evaluate f at each critical number in the interval 3) Evaluate each endpoint of the interval 4) The least of these values is the minimum, the most is the maximum 1)
![Example Find the absolute maximum and minimum of on the interval [ -1/2, 4]](http://slidetodoc.com/presentation_image_h/f64ad031f3f07d9bd49eb526ae161124/image-10.jpg "Example Find the absolute maximum and minimum of on the interval [ -1/2, 4]")
Example Find the absolute maximum and minimum of on the interval [ -1/2, 4]
Practice: Find Max and Min
The Hubble Space Telescope was deployed by the Space Shuttle. A model for the velocity of the shuttle during the mission from liftoff at t=0 to when the boosters were jettisoned at t=126 s is given by Use this model to estimate the maximum and minimum values of acceleration.
- Slides: 12