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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.

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

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

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

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

11 step curve sketch

Intercepts

Symmetry

Domain and Range

Continuous

Vertical asymptotes

Differentiability

Extrema

Concavity

Points of Inflection

Horizontal Asymptotes

Slant Asymptotes

Symmetry

Domain and Range

Continuous

Vertical asymptotes

Differentiability

Extrema

Concavity

Points of Inflection

Horizontal Asymptotes

Slant Asymptotes