MAT 1300 Lecture Notes - Lecture 9: Maxima And Minima, Minimax
9 Oct 2018
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MAT 1300 LECTURE 9- GLOBAL MAXIMUM AND MINIMUM & CURVE SKETCHING
Theorem
If f is continuous on [a,b] then f has a global max and a global min somewhere in [a,b]
Global max/min can occur only either at critical points or the end points
Example f(x) =x^3 -3x^2-9x +5
Find the global max and the global min on[-2,6]
SOLUTION
f’(x) =3x^2 -6x-9
=3(x^2 -2x-3)
=3(x-3)(x+1)
The critical numbers are x=3 and x= -1
Critical numbers
x
f(x)
3
-22
-1
10
Given numbers
x
f(x)
-2
3
6
59
- F has a global maximum at x=6, the global max of f on[-2,6] is 59
- F has a global minimum at x=3, the global max of f on[-2,6] is -22
THE ONLY CRITICAL POINT IN TOWN TEST
Suppose f is a continuous function with only one critical number c then:
f(c ) is a local max/min → f(c ) is a global max /min
Example
f(x ) = x- e^x
Find the global max and global min of f on (-infinity , +infinity)
Solution
f’(x) = 1- e^x
f’(x) =0 only when x=0
So x=0 is the only critical number
f” (x) = -e^x
f”(0) = -1 <0
Document Summary
Mat 1300 lecture 9- global maximum and minimum & curve sketching. If f is continuous on [a,b] then f has a global max and a global min somewhere in [a,b] Global max/min can occur only either at critical points or the end points. Find the global max and the global min on[-2,6] The critical numbers are x=3 and x= -1 x. F has a global maximum at x=6, the global max of f on[-2,6] is 59. F has a global minimum at x=3, the global max of f on[-2,6] is -22. Suppose f is a continuous function with only one critical number c then: f(c ) is a local max/min f(c ) is a global max /min. Find the global max and global min of f on (-infinity , +infinity) Solution f"(x) = 1- e^x f"(x) =0 only when x=0. So x=0 is the only critical number f (x) = -e^x f (0) = -1 <0.
