Section_2.3-Getting Information from the Graph of a Function

Section_2.3-Getting - 2 The function f x = x 2 x ∈-∞ 0 ∪(0 ∞ has no local minimum or maximum 3 The functions f x = x x 3 x 5 have no local

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Section 2.3 Getting Information from the Graph of a Function Increasing and Decreasing Functions DEFINITION: A function f is called increasing on an interval I if f ( x 1 ) < f ( x 2 ) whenever x 1 < x 2 in I It is called decreasing on an I if f ( x 1 ) > f ( x 2 ) whenever x 1 < x 2 in I EXAMPLES: 1. The function f ( x ) = x 2 is decreasing on ( -∞ , 0) and increasing on (0 , ) . 2. The function f ( x ) = x 3 is increasing everywhere, that is on ( -∞ , ) . 3. The function f ( x ) = 1 x is decreasing on ( -∞ , 0) and on (0 , ) . 1
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Local Maximum and Minimum Values of a Function DEFINITION: A function f has a local maximum (or relative maximum ) at c if f ( c ) f ( x ) when x is near c. [This means that f ( c ) f ( x ) for all x in some open interval containing c. ] Similarly, f has a local minimum at c if f ( c ) f ( x ) when x is near c. EXAMPLES: 1. The function f ( x ) = x 2 has the local minimum at x = 0 and has no local maximum.
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Unformatted text preview: 2. The function f ( x ) = x 2 , x ∈ (-∞ , 0) ∪ (0 , ∞ ) has no local minimum or maximum. 3. The functions f ( x ) = x, x 3 , x 5 have no local minimum or maximum. 4. The functions f ( x ) = sin x, cos x, sec x, csc x have infinitely many local minimums and maximums. The functions f ( x ) = tan x, cot x have no local minimums or maximums. 5. The function f ( x ) = x 4 + x 3-11 x 2-9 x + 18 = ( x-3)( x-1)( x + 2)( x + 3) has two local minimums at x ≈ -2 . 6 and x ≈ 2 . 2 and a local maximum at x ≈ -. 4 . 6. The function f ( x ) = 1 has a local minimum and maximum at any point on the number line. 2...
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This note was uploaded on 02/26/2012 for the course MATH 8650 taught by Professor Kiryltsishchanka during the Spring '12 term at NYU.

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Section_2.3-Getting - 2 The function f x = x 2 x ∈-∞ 0 ∪(0 ∞ has no local minimum or maximum 3 The functions f x = x x 3 x 5 have no local

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