Difference between revisions of "Extrema"
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'''Maxima''' are a type of extrema or critical number for functions, which can be found with calculus. | '''Maxima''' are a type of extrema or critical number for functions, which can be found with calculus. | ||
| − | The ''absolute'' maximum of a function on an interval is either its value at an end point or its value when its first derivative is zero and its second derivative is less than zero. | + | The ''absolute'' maximum of a function is the point in the domain at which the function achieves its largest value. For example, the absolute maximum of <math>f(x) = 1- x^2</math> occurs at <math>x=0</math>, where <math>f(0) = 1</math>. |
| + | A useful fact from calculus is that the maximum of a function on an interval is either its value at an end point or its value when its first derivative is zero and its second derivative is less than zero. | ||
| − | The ''relative'' maximum of a function is when its first derivative is zero and its second derivative is less than zero or, alternatively, its first derivative is positive approaching the point from the left and negative to its right. | + | The ''relative'' maximum of a function is a point which at which the function takes on its largest value on some neighborhood of the point, i.e., the function takes on smaller values at all sufficiently close points. For example, the function <math>f(x) = e^{-x^2} \cos x</math> has many relative maxima, and a unique absolute maximum at <math>x=0</math>. A relative maximum of a function can occur when its first derivative is zero and its second derivative is less than zero or, alternatively, its first derivative is positive approaching the point from the left and negative to its right. |
To find the '''maxima''' of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero. | To find the '''maxima''' of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero. | ||
[[Category:mathematics]] | [[Category:mathematics]] | ||
[[Category:calculus]] | [[Category:calculus]] | ||
Revision as of 03:51, December 25, 2009
Maxima are a type of extrema or critical number for functions, which can be found with calculus.
The absolute maximum of a function is the point in the domain at which the function achieves its largest value. For example, the absolute maximum of 
occurs at 
, where 
.
A useful fact from calculus is that the maximum of a function on an interval is either its value at an end point or its value when its first derivative is zero and its second derivative is less than zero.
The relative maximum of a function is a point which at which the function takes on its largest value on some neighborhood of the point, i.e., the function takes on smaller values at all sufficiently close points. For example, the function 
has many relative maxima, and a unique absolute maximum at . A relative maximum of a function can occur when its first derivative is zero and its second derivative is less than zero or, alternatively, its first derivative is positive approaching the point from the left and negative to its right.
To find the maxima of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero.
