Sect. 4.2
What is the Extreme Value Theorem?
If f is continuous on a closed interval, then f has both a minimum and a maximum on the interval.
Definition of critical numbers.
What is f'(c) = 0 or if f(x) is not differentiable at c, c is a critical number?
Concavity
All of its tangents lie below the curve on the interval. The slope of the tangent lines are increasing.
What is concave upward?
Relating f and f'
f'(x) > 0
means...
f(x) increases
Sarah says there is an absolute maximum at an endpoint. May this occur?
What is yes, the absolute max/min of a function may occur at the endpoints.
Critical Numbers only exist if the number line test shows that it changes sign at c.
What is false?
(This is the case for inflection points.)
Determine the points of inflection.
f(x) = x4 - 4x3
What is x = 0 and x = 2?
Graph f(x) = 2x3 - 3x2 - 12x + 1
To find the max and min of f on [a, b]:
Find the critical numbers of f(x) in (a, b)
Evaluate f(x) at each critical number in (a, b).
The least of these values is the minimum. The greatest is the maximum.
What is missing?
Evaluate f(x) at each endpoint in [a, b].
If the sign changes from negative to positive at c, it is a relative max.
What is false?
(- \ to / +)
If f''(c) = 0, does it pass the Second Derivative Test?
What is no?
Find the absolute maximum and/or minimum of sinx -sinx2, on [0, 2pi].
The absolute max is 1/4 when x = π/6, 5π/6.
The absolute min is –2 when x =3π/2.
Find relative max or min of
f(x) = x3 - 9x2 + 24x
Rel max at x = 2
Rel min at x = 4
Use the 2nd derivative test to find the local max and/or min for ex - 5x.
What is a local minimum at x = ln5?
Rayla was given a problem. She believes she made a mistake somewhere but is unsure. Did she make a mistake? If so what is it?
Find the absolute extrema, if they exist, for the function
f(x) = x^2/(x-1) on [-2,3]
f’(x) = {2x(x-1) - x^2}/(x-1)^2
f’(x) = (x^2 - 2x)/(x-1)^2
f’(x) = {x(x-2)}/(x-1)^2
Critical Numbers: x = 1, x = 0, and x = 2
x = 1 can not be a critical number as it is not in the domain
Relative max is (-1, cube root of 2)
Relative min is (1, -cube root of 2)
Use the 2nd derivative test to find the local max and/or min for f(x) = 10x6 - 24x5 + 15x4
What is a local min at x = 0?