Extrema and Differentiation Quiz

What is the main focus of calculus in determining the behavior of a function on an interval?

• Identifying the discontinuities of the function

• Finding the domain of the function

• Calculating the average value of the function

• Determining the extrema of the function

In Figure 3.1(a) and (b), the function f(x) = x^2 + 1 has both a minimum and a maximum on which type of interval?

• Open interval

• Closed interval

• Half-open interval

• Infinite interval

What can affect the existence of an extremum on an interval?

• The concavity of the function

• The slope of the function at the extremum

• The continuity of the function

• The value of the function at the extremum

How can a relative maximum be informally described for a continuous function?

• As occurring at the intersection of two functions

• As occurring at the origin of the graph

• As occurring on a 'hill' on the graph

• As occurring on a straight line on the graph

What are the two ways in which a hill and valley can occur in a graph?

• Straight and curved, wavy and zigzag

• Smooth and rounded, sharp and peaked

• Increasing and decreasing, constant and variable

• Positive and negative, even and odd

At each relative extremum, the derivative of the function is either zero or does not exist. What are these special points called?

• Stationary points

• Inflection points

• Critical points

• Singular points

What is the value of the derivative at the point (3, 2) shown in Figure 3.3(a)?

• f'(3) = 0

• f'(3) = undefined

• f'(3) = 1

• f'(3) = -1

At x = 0, the derivative of f(x) = |x| does not exist because the following one-sided limits differ. What are these limits?

• lim(x->0-) |x| = 0, lim(x->0+) |x| = 1

• lim(x->0-) |x| = 1, lim(x->0+) |x| = 0

• lim(x->0-) |x| = 1, lim(x->0+) |x| = -1

• lim(x->0-) |x| = -1, lim(x->0+) |x| = 1

What are the critical numbers of the function f(x) = sin x?

• x = 0, x = π

• x = π/2, x = 3π/2

• x = π/4, x = 3π/4

• x = π/6, x = 5π/6

What are the extrema of f(x) = 3x^4 – 4x^3 on the interval [–1, 2]?

• Maximum at x = 0, minimum at x = -1

• Maximum at x = 1, minimum at x = 0

• Maximum at x = -1, minimum at x = 2

• Maximum at x = 2, minimum at x = 1