Applications of the First and Second Derivative by stariya

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```									                APPLICATIONS OF THE FIRST AND SECOND DERIVATIVE

=0                             >0                                   <0
f (x)     Critical Point(s)             f (x) increasing                    f (x) decreasing

f (x)   Inflection Point(s)         f(x) Concave up                    f(x) Concave down
Relative Minimum(s)                 Relative Maximum(s)

Notes:

 For polynomial functions there are at most n – 1 relative minima + relative maxima
 For polynomial functions there are at most n – 2 inflection points
 Increasing/decreasing Intervals are determined by the critical points and using test points for f (x)
 Concavity Intervals are determined by the inflection points and using test points for f (x)

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APPLICATIONS OF THE FIRST AND SECOND DERIVATIVE
Example I: f ( x)  2 x - x  3 has at most 2 relative maxima + relative minima and at most 1 inflection point
3     2

f ( x)  6 x 2 - 2x      f ( x)  12x - 2
1. Critical points at x=0 and x = 1/3
2. Inflection point at x = 1/6
3. f(x) increasing on - ,0  & (1 / 3, )
f(x) decreasing on 0,1/3 
4. f(x) Concave down on - ,1/6 
Relative maximum at x = 0
f(x) Concave up on 1/6,  
Relative minimum x = 1/3

Example II: f ( x)  x has at most 2 relative maxima + relative minima and at most 1 inflection point
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f ( x)  3x 2       f ( x)  6 x
1. Possible critical point at x=0
2. Inflection point at x = 0
3. f (x) is positive everywhere therefore
f(x) is Increasing on - ,  
4. Since f(x) is always increasing there is no critical
point, no concavity, and no relative max or min.

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APPLICATIONS OF THE FIRST AND SECOND DERIVATIVE

Example III: f ( x)  2 x  3 x  3 has at most 1 relative maxima + relative minima and no inflection point
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f ( x)  4 x  3   f ( x)  4
1. Possible critical point at x=3/4
2. -4≠0 therefore no Inflection point
3. f(x) is Increasing on - ,3/4 
f(x) is decreasing on 3/4,  
4. Since f (x) is always negative f(x) is concave down
everywhere - ,   and has a relative max at x=3/4

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