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|Dated: Jul 09, 13|
Announcement for GDB Topic:
Discuss the applications of "Maxima and Minima Value of Functions" in various fields of life.
Opening Date: July 15, 2013 at 12:01 AM
Closing Date: July 16, 2013 at 11:59 PM
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The y-value f(c) is a local maximum value (also called relative maximum value) of f if there is an open interval containing the x-value c. When the graph of the function f, continuous at x=c, is increasing on the immediate left of the number x=c and decreasing on the immediate right of the number x=c, then the value of f at c is locally the largest, i.e., f(c) is a local maximum. This test can be extended to endpoints in the domain of f: if x=c is a left endpoint and decreasing on the immediate right, or if x=c is a right endpoint and increasing on the immediate left, then f(c) is a local maximum.
The y-value f(c) is a local minimum value (also called relative minimum value) of f if there is an open interval containing the x-value c. When the graph of the function f, continuous at x=c, is decreasing on the immediate left of the number x=c and increasing on the immediate right of the number x=c, then the value of f at c is locally the smallest, i.e., f(c) is a local minimum. This test can be extended to endpoints in the domain of f: if x=c is a left endpoint and increasing on the immediate right, or if x=c is a right endpoint and decreasing on the immediate left, then f(c) is a local minimum.
Determining Local Maxima and Local Minima
Any value of x in the domain of f is called a critical number of f' (also called critical point or critical value) if either f'(x)=0 or f'(x) does not exist. For continuous functions, the local maxima and local minima can only occur at the critical numbers or endpoints of the domain of f. These numbers separate the domain of f into intervals. At each critical number or endpoint, there are three possibilities.
If the sign of f'(c) is positive on the left side (interval to the left) of a critical number of f' and negative on the right side, suggesting visually that the function is increasing on the left side and decreasing on the right side, then f has a local maximum at that critical number. If the sign of f'(c) is positive on the left side of a right endpoint or negative on the right side of a left endpoint, then f has a local maximum at that endpoint.
If the sign of f'(c) is negative on the left side of a critical number of f' and positive on the right side, suggesting visually that the function is decreasing on the left side and increasing on the right side, then f has a local minimum at that critical number. If the sign of f'(c) is negative on the left side of a right endpoint or negative on the right side of a left endpoint, then f has a local minimum at that endpoint.
If there is no change in sign of f'(c) from either side of a critical number to the other side, then the critical number is not a local maximum or local minimum for f. The curve has a horizontal tangent at the critical number but the point of tangency is not a turning point.
Function can have “hills and valleys” places where they reach a maximum and minimum values. It may not be the minimum or maximum for the whole function. I our daily life we saw many hills and valleys. The hills are maximum point and the valleys are the minimum point. The plural of minimum is Minima and the plural of maximum is Maxima. Maxima and Minima are collectively called Extrema.The process of a finding maximum or minimum values is called optimization. Maximum or minimum can be defined as largest and smallest of function at a given point in it Domain or outside its domain. Local maxima is the Point that is at the peak with respect to the local surrounding points. Similarly, local minima can be defined as the lowest point with respect to the surrounding points. Maxima or minima can be many points as they are considered locally in the graph. It also resembles with the Set theory in a context that we have least and peak points in the set. In our life the Government or an industry regulatory can set a maximum price in an attempt to prevent the market price from rising above a certain level . The minimum price is a price below which the market price cannot fall.
, any applications of maxima and minima are purely graphical:
If you are taking ECON 101 as an elective course this term, you are likely familiar with the many applications of maxima and minima in the world of economics. Maxima and minima are used to maximize beneficial values (profit, efficiency, output, etc.) and to minimize negative ones (expenses, effort, etc.).
Applications: "Real World"
Maxima and minima pop up all over the place in our daily lives. They can be found anywhere we are interested in the highest and/or lowest value of a given system; if you look hard enough, you can probably find them just about anywhere! Here are just a few examples of where you might encounter maxima and minima:
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One of the great powers of calculus is in the determination of the maximum or minimum value of a function. Take f(x) to be a function of x. Then the value of x for which the derivative of f(x) with respect to x is equal to zero corresponds to a maximum, a minimum or an inflexion point of the function f(x).
For example, the height of a projectile that is fired straight up is given by the motion equation:
Taking y0 = 0, a graph of the height y(t) is shown below.
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function y(t) plotted as a function of t. The derivative is positive when a function is increasing toward a maximum, zero (horizontal) at the maximum, and negative just after the maximum. The second derivative is the rate of change of the derivative, and it is negative for the process described above since the first derivative (slope) is always getting smaller. The second derivative is always negative for a "hump" in the function, corresponding to a maximum