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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2:29 minutes

Problem 4.3.68a

Textbook Question

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).

Verified step by step guidance

Start by understanding the behavior of the function around the local minimum and maximum. A local minimum at (1, 1) means that the derivative changes from negative to positive at x = 1, indicating a 'valley' in the graph. Similarly, a local maximum at (3, 3) means the derivative changes from positive to negative at x = 3, indicating a 'peak'.

Consider the general shape of the graph. Between x = 1 and x = 3, the function should increase to reach the local maximum at (3, 3). Before x = 1 and after x = 3, the function should decrease, reflecting the local minimum and maximum respectively.

Sketch the graph starting from the left of x = 1. The function should decrease until it reaches the point (1, 1). At this point, the slope of the tangent line is zero, indicating a local minimum.

From x = 1 to x = 3, sketch the graph such that it increases to reach the local maximum at (3, 3). At this point, the slope of the tangent line is zero again, indicating a local maximum.

Finally, sketch the graph beyond x = 3, where the function should decrease again. Ensure the graph reflects the change in slope at the local minimum and maximum points, creating a smooth curve that is differentiable everywhere.

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Key Concepts

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

Differentiable Function

A differentiable function is one that has a derivative at every point in its domain. This means the function is smooth and continuous, without any sharp corners or cusps. Understanding differentiability is crucial for sketching graphs, as it ensures the function's behavior can be predicted using its derivative.

Local Minimum and Maximum

A local minimum is a point where the function value is lower than all nearby points, while a local maximum is where the function value is higher than all nearby points. These points are critical in graph sketching, as they indicate where the function changes direction, often corresponding to where the derivative equals zero.

Critical Points and Derivatives

Critical points occur where the derivative of a function is zero or undefined, indicating potential local minima or maxima. Analyzing the derivative helps determine the nature of these points, guiding the sketching of the function's graph by showing where it increases or decreases.

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Related Practice

Textbook Question

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) = x − 6√(x − 1)

Textbook Question

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) = (x² − 3) / (x − 2), x ≠ 2

Textbook Question

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) = x¹ᐟ³(x + 8)

Textbook Question

Theory and Examples

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

Let f(x) = |x³ − 9x|.

d. Determine all extrema of f.

Textbook Question

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).

Textbook Question

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local maxima at (1, 1) and (3, 3)

Textbook Question

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local minima at (1, 1) and (3, 3).

Textbook Question

Identifying Extrema

In Exercises 15–18:

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

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

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Theory and ExamplesSketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).

Key Concepts

Differentiable Function

Local Minimum and Maximum

Critical Points and Derivatives

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Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).