5. Graphical Applications of Derivatives

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

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y = √𝓍² ― 1

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

__________

y = √ 3 + 2𝓍 ―𝓍²

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)

Estimate the open intervals on which the function y = ƒ(𝓍) is

a. increasing.

b. decreasing.

c. Use the given graph of ƒ' to indicate where any local extreme

values of the function occur, and whether each extreme

is a relative maximum or minimum.

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Theory and Examples

In Exercises 51–54,

c. For what values of x, if any, is f' positive? Zero? Negative?

y = −x²

Theory and Examples

In Exercises 51–54,

d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)

y = −1/x

37. What value of a makes f(x) = x^2 +(a/x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?

38. What values of a and b make f(x) = x^3 + ax^2 + bx have

b. a local minimum at x = 4 and a point of inflection at x = 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) = (x + 1)², −∞ < x ≤ 0

Identifying Extrema

In Exercises 41–52:

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

g(x) = x² − 4x + 4, 1 ≤ x < ∞

Identifying Extrema

In Exercises 41–52:

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

g(x) = −x² − 6x − 9,−4 ≤ x < ∞

Identifying Extrema

In Exercises 41–52:

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

f(t) = 12t − t³, −3 ≤ t < ∞

Identifying Extrema

In Exercises 41–52:

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

f(t) = t³ − 3t², −∞ < t ≤ 3

Identifying Extrema

In Exercises 41–52:

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

k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0

Identifying Extrema

In Exercises 41–52:

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

g(x) = (x − 2) / (x²−1), 0 ≤ x < 1