These are the two values of the first derivative at a critical number.
What is zero or does not exist?
100
Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then this is guaranteed to exist for at least one number c in (a, b).
What is f'(c) = 0?
100
This is what both the First and Second Derivative Tests are locating.
What are local (relative) extrema?
100
These are the characteristics of a function on an interval when f ' (x) is positive and f '' (x) is negative.
What is increasing and concave down?
100
Along with identifying all given quantities and quantities to be determined, this is part of the first step in solving optimization problems.
What is draw a picture?
200
This is what occurs at a point of inflection.
What is the change from concave up to concave down or vice versa?
200
The number of zeros (roots) of f(x)=x^3+4x+1 on the interval [-1, 1].
What is one?
200
These numbers are required to find local extrema using both the First and Second Derivative Tests.
What are critical numbers?
200
This characteristic of f(x) is true when f ' (x) is increasing.
What is concave up?
200
The function you are attempting to optimize is called this.
What is the objective function?
300
These are the two possible names for the point (x, f(x)) when x = c is a critical number.
What is a stationary point or a point of nondifferentiability?
300
If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that this is true.
What is f'(c) = [f(b) - f(a)]/[b - a].
300
These are the three possible results of the Second Derivative test and their corresponding types of extrema.
What is a positive result meaning local minimum, a negative result meaning local maximum, and zero meaning test fails and must use First Derivative Test to find extrema?
300
These are the equations of the horizontal asymptotes for the function f(x) = sqrt(4x^2+1)/(x+1).
What are y=2 and y= -2 ?
300
This is the main reason for finding a constraint equation.
What is to rewrite the objective function in terms of a single variable?
400
These are the critical numbers of f(x) = x^2(x^2 - 4).
What are 0, 2, -2?
400
Let f(x) = x^4 - 2x^2. These are the values of c in the interval (-2, 2) guarenteed by Rolle's Theorem.
What are 0, 1, and -1?
400
These are the local extrema of f(x) = (x^2 - 4)^2/3.
What is a relative minimum at (-2, 0) and (2, 0) and a relative maximum at (0, 16^1/3)?
400
The interval(s) on which y = x^3 - 12x is increasing.
What is (-infinity, -2) U (2, infinity)?
400
This is the point on f(x) = x^2 that is closest to (2, 1/2).
What is (1, 1)?
500
These are the points of inflection of f(x) = x^4 - 4x^3
What are (0,0) and (2,-16)?
500
Let f(x) = x^(2/3) on [0, 1]. This is the value of c that is guarenteed by the Mean Value Theorem.
What is 8/27?
500
These are the relative extrema for f(x) = -3x^5 + 5x^3.
What is a local minimum at (-1, -2) and a local maximum at (1, 2)?
500
This is the interval on which f(x)=2x^2/(x^2-1) is concave down, given that f '' (x) = (12x^2+4)/(x^2-1)^3.
What is (-1,1)?
500
A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd and does not need fencing along the river. These are the dimensions required to use the least amount of fencing.