Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Intro to Extrema

Calculus

5. Graphical Applications of Derivatives

Intro to Extrema

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2:47 minutes

Problem 3.2.51c

Textbook Question

Theory and Examples

In Exercises 51–54,

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

y = −x²

Verified step by step guidance

First, find the derivative of the function y = -x². The derivative, denoted as f', represents the rate of change of the function. Use the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1).

Apply the power rule to y = -x². The derivative of -x² is f'(x) = -2x. This is because the derivative of x² is 2x, and the negative sign remains as a constant factor.

To determine where f' is positive, set the derivative -2x > 0 and solve for x. This inequality will tell you the range of x values where the function is increasing.

To find where f' is zero, set the derivative -2x = 0 and solve for x. This will give you the x value where the function has a horizontal tangent, indicating a potential maximum or minimum point.

To determine where f' is negative, set the derivative -2x < 0 and solve for x. This inequality will tell you the range of x values where the function is decreasing.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function, denoted as f', represents the rate of change of the function with respect to its variable. It provides information about the slope of the tangent line to the curve at any given point. For the function y = -x², the derivative helps determine where the function is increasing, decreasing, or has a horizontal tangent.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are important for identifying where a function changes from increasing to decreasing or vice versa. For y = -x², finding the critical points involves setting the derivative equal to zero and solving for x, which helps in analyzing the behavior of the function.

Sign of the Derivative

The sign of the derivative indicates whether a function is increasing or decreasing. If f' is positive, the function is increasing; if f' is negative, the function is decreasing; and if f' is zero, the function has a horizontal tangent. Analyzing the sign of the derivative for y = -x² helps determine the intervals where the function is increasing, decreasing, or constant.

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Related Practice

Textbook Question

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

______

y = √𝓍² ― 1

Textbook Question

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

Textbook Question

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)

Textbook Question

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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Textbook Question

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

Textbook Question

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?

Textbook Question

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?

Textbook Question

Identifying Extrema

In Exercises 41–52:

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