Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


y = x − 3x²ᐟ³

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

b. On what open intervals is f increasing or decreasing?

f′(x) = (x − 1)(x + 2)(x − 3)

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

a. What are the critical points of f?

f′(x) = 1− 4/x², x ≠ 0

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

b. On what open intervals is f increasing or decreasing?

f′(x) = 1− 4/x², x ≠ 0

Identifying Extrema

In Exercises 19–40:

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

f(r) = 3r³ + 16r

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

g(x) = √(2x − x²)

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.