Graphing Using First And Second Derivatives - UC Davis Math

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1. If the first derivative f' is positive (+) , then the function f is increasing ($ \uparrow $) .

2. If the first derivative f' is negative (-) , then the function f is decreasing ( $ \downarrow $) .

3. If the second derivative f'' is positive (+) , then the function f is concave up ($ \cup $) .

4. If the second derivative f'' is negative (-) , then the function f is concave down ($ \cap $) .

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 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 .

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