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Section 4.3

# Section 4.3 - range of f ◦ 7 The point x=a determines an...

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Section 4.3: How Derivatives Affect the Shape of a Graph 1. If the first derivative f' is positive (+) , then the function f is increasing ( ) . 2. If the first derivative f' is negative (-) , then the function f is decreasing ( ). 3. If the second derivative f'' is positive (+) , then the function f is concave up ( ). 4. If the second derivative f'' is negative (-) , then the function f is concave down ( ). 5. The point x=a determines a relative maximum for function f if f is continuous at x=a , and the first derivative f' is positive (+) for x<a and negative (-) for x>a . The point x=a determines an absolute maximum for function f if it corresponds to the largest y-value in the range of f. 6. The point x=a determines a relative minimum for function f if f is continuous at x=a , and the first derivative f' is negative (-) for x<a and positive (+) for x>a . The point x=a determines an absolute minimum for function f if it corresponds to the smallest y-value in the

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Unformatted text preview: range of f. ◦ 7. The point x=a determines an inflection point for function f if f is continuous at x=a , and the second derivative f'' is negative (-) for x<a and positive (+) for x>a , or if f'' is positive (+) for x<a and negative (-) for x>a. ◦ 8. THE SECOND DERIVATIVE TEST FOR EXTREMA (This can be used in place of statements 5. and 6.) : Assume that y=f(x) is a twice-differentiable function with f'(c)=0 . ◦ a.) If f''(c)<0 then f has a relative maximum value at x=c . ◦ b.) If f''(c)>0 then f has a relative minimum value at x=c . Example: f(x) = x 3- 3x 2 = 3x (x - 2) = 0 x=2, x=0 f''(x) = 6x - 6 = 6 (x - 1) = 0 FROM f' : f is ( ) for x<0 and x>2 ; f is ( ) for 0<x<2 ; f has a relative maximum at x=0 , y=0 ; f has a relative minimum at x=2 , y=-4 . FROM f'' : f is ( ∪ ) for x>1 ; f is ( ∩ ) for x<1 ; f has an inflection point at x=1 , y=-2 ....
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