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### Critical Number

let f be defined at c. If f '(c)=0 or if f is not differentiable at c, then c is a critical number of f

### The First Derivative Test

Let c be a critical number of a function f which is continuous on the open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows

1. If f '(x) changes from negative to positive at c then f has a relative minimum at (c,f(c))

2. If f '(x) changes from positive to negative at c then f had a relative maximum at (c,f(c))

3. If f '(x) is both positive or both negative around c then f(c) is neither a relative minimum or relative maximum.

### Point of Inflection

If (c,f(c)) is a point of inflection of f, than either f "(c)=0 or f "(c) does not exist. Concavity must change at c though.

### Second Derivative Test

Let f be a function such that f '(c)=0 and the second derivative of f exists on the open interval containing c

1. If f "(c)>0 then f has a relative minimum at (c,f(c))

2. If f "(c)<0 then f has a relative maximum at (c,f(c))

3. If f "(c)=0 then the test fails. f may have a relative maximum or a relative minimum or neither. In such cases, use the First derivative Test

### Rolle's Theorem

If

1. f(x) is continuous on the interval [a,b]

2. f(x) is differentiable on the interval (a,b)

3. f(a) = f(b)

then Rolle's Theorem can be applied

### Mean Value Theorem

If

1. f(x) is continuous on the interval [a,b]

2. f(x) is differentiable on the interval (a,b)

then the statement f '(c) = [f(b) - f(a)] / [b-a] is true