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Definition


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Four ways in which a function can fail to be differentiable at a point 

Definition
• Discontinuity • Corner/Point • Cusp • Vertical tangent 


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To justify a critical number: 

Definition
f'(x)=0 or does not exist 


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Justify f(x) is increasing. 

Definition


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Justify f(x) is decreasing.


Definition


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Justify f(x) is concave up.


Definition
f"(x)>0 or f'(x) is increasing.



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Justify f(x) is concave down.


Definition
f"(x)<0 or f'(x) is decreasing. 


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Definition


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Definition


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Definition


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Definition
f'(x)=0 or undefined and changes from positive to negative at x. 


Term

Definition
f'(x)=0 or undefined and changes from negative to positive at x. 


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Intermediate Value Theorem: If f is continous on the interval [a,b], then... 

Definition
f must take on every value (height) from f(a) to f(b). 


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If f is differentable at a point, then... 

Definition
...it must be continuous at that point. (Differentiability implies continuity.) 


Term

Definition
f"(x)=0 or undefined
AND changes sign 

