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Critical Numbers
Values of x in the domain of f (x) where f 'x= 0 or f 'x is undefined

How do you find absolute minimums and maximums on a closed interval?
 1. Find critical numbers
 2. Find values for the function at each endpoint and critical number
 3. Determine absolute max and absolute min for the interval by your results from step 2.

Rolle’s Theorem
 If f is differentiable on the interval (a , b) and f(a) = f(b) then there is at least one place in the interval
 where f 'x= 0

Mean Value Theorem (MVT)
 If f (x) is continuous and differentiable on the interval [a , b], there is someplace in the interval where f ' x
 equals the slope of the line going through the endpoints of the interval.

First Derivative ITSC
1) Use critical numbers to determine intervals.
 Interval
 Test Value
 Sign of f ' x
 Conclusion

If sign of f ' x is positive what is happening to f(x)?
f(x) is increasing

If sign of f ' x is negative then f(x) is _______.
f(x) is decreasing

First Derivative Test
Do a first derivative ITSC,
 If f ' x is changing from pos to neg, then relative max
 If f ' x is changing from neg to pos, then relative min

Concave Up
f'(x) is increasing, f''(x)is positive, f(x) will hold water

Concave Down
f ' x is decreasing, f '' x is negative, f (x) will shed water

Second Derivative ITSC
Find where the 2nd Derivative is 0 or undefined. Use those numbers to make intervals
 Intervals
 Test Value
 Sign of f '' x
 Conclusio

If f '' x is positive
interval is concave up

If f '' x is negative,
interval is concave down

Inflection Point
 Where the function is switching from concave up to concave down or
 from concave down to concave up.

2nd Derivative Test
 1. Plug your critical numbers into the 2nd Derivative
 2. If the result is positive, then the there is a relative min at the critical
 number
 3. If the result is negative, then there is a relative max at the critical
 number.
 4. If the result is 0, the test fails




Definition of a Differential
dy= f(x)dx

Definition of Δy
Δy= f(x+Δx) − f(x)

Optimization Problem
 1. Write your two equations. The equation that you are minimizing or maximizing is the Primary Equation.
 2. Get your Primary equation in terms of one variable by substituting in the other equation.
 3. Differentiate your primary equation and find the critical numbers.
 4. Show that the critical numbers are a max or min by the 2nd Derivative Test or a First Derivative ITSC
 5. Answer the question.

