University:
California State University, Northridge
Course:
MATH 150AL | Calculus I Laboratory
Academic year:

2024
Views:

392

Pages:

1
Author:

podalekomy26

0.3 Local and Global Extrema Example: An extremum of a function is either its maximum or its minimum. A point x0 is called a relative maximum of the function f if f (x0 ) ≥ f (x) for all values of x close to x0 . I.e. the value of the function at x0 is larger than at neighboring points. At a local minimum the value of the function £ √ ¤ is smaller than at neighboring points. Let f (x) = sin x2 on the interval 0, 2π . Produce a graph of this function. This function has 2 local minima and 2 local maxima. The candidates for these local extrema are the end points of the √ ¤ and the points where the £ interval graph has a horizontal tangent line. Since 0, 2π is a closed interval, the end points are included. To find the local extrema we first find the critical points, i.e. the points where the derivative vanishes or does not exist. To do this we solve √ √ ª © √ √ ª ¡ ¢ © f 0 (x) = 2 cos x2 x = 0, Solution is: x = 21 2 π , x = − 12 2 π , {x = 0} . £ √ ¤ The computer returns 3 solutions, only 2 of which are in the interval 0, 2π .However, q 3π it does not find the third solution in the interval at x = 2 . From this and the = 0) and q we observe that f has local minima at√x√= 0 (f (0) q graph √ √ 1 1 3π 3π ) = −1) and local maxima at x = 2 2 π(f ( 2 2 π) = 1) x= 2 (f ( √ 2 √ and at x = 2π(f ( 2π) = 0). A function has a global maximum (or minimum) at x0 in an interval [a, b] , if f (x0 ) ≥ f (x) (or f (x0 ) ≤ f (x)) for all a ≤ x ≤ b. A continuous function in a closed interval has always a global maximum and a global minimum. In this case (the closed interval) the global maximum is the largest local maximum, the global minimum is the smallest √ √ local minimum. √ √In our example the largest local maximum is at x = 21 2 π , since f ( 21 2 π) = 1 > 0 = q q √ 3π f ( 2π). The smallest local minimum is at x = 3π , where f ( 2 ) = −1. √2 If our interval of interest would have been [0, π) ,then the function would have a local and global minimum, but neither a local nor a global maximum.

Local and Global Extrema

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