Theory and Examples


Cubic functions Consider the cubic function f(x) = ax³ + bx² + cx + d.


b. How many local extreme values can f have?

Theory and Examples

In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.

y = 3x + tan x

Graph f(x) = 2x^4 -4x^2 + 1 and its first two derivatives together. Comment on the behavior of f in relation to the signs and values of f' and f".

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = (x − 2)²ᐟ³.

b. Show that the only local extreme value of f occurs at x = 2.

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

b. Does f'(-3) exist?

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

g(x) = x√8 − x²

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

f(x) = x³ / (3x² + 1)

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

f(x) = √(x² − 2x − 3), 3 ≤ x < ∞

Sketch the graph of a twice-differentiable function y=f(x) that passes through the points (-2,2), (-1,1), (0,0),(1,1), and (2,2) and whose first two derivatives have the following sign patterns.