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 68d

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).

Verified step by step guidance

Start by understanding the concept of local minima. A local minimum is a point where the function value is lower than all nearby points. This means the derivative at these points is zero, indicating a horizontal tangent line.

Identify the given local minima points: (1, 1) and (3, 3). These points suggest that the function has a 'valley' or 'dip' at these coordinates.

Consider the behavior of the function around these points. Since the function is differentiable, it is smooth and continuous. The derivative changes sign around these minima points, from negative to positive.

Sketch the graph by plotting the points (1, 1) and (3, 3) on a coordinate plane. Draw a curve that dips down at these points, ensuring the tangent is horizontal at these minima.

Ensure the graph is smooth and continuous, reflecting the differentiability of the function. Between the minima, the graph should rise, and outside these points, it should fall, creating a 'U' shape around each minimum.

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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 Minima

A local minima of a function is a point where the function value is lower than all other nearby points. At a local minima, the derivative of the function changes from negative to positive, indicating a 'valley' in the graph. Identifying local minima is essential for sketching the graph accurately, as it highlights where the function reaches its lowest points locally.

Graph Sketching

Graph sketching involves plotting the general shape of a function based on its critical points, such as local minima and maxima, and its behavior at infinity. It requires understanding the function's derivative to determine where the function is increasing or decreasing, and how it curves. This skill is vital for visualizing the function's behavior and ensuring the graph reflects all given conditions.

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Related Practice

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 minimum at (1, 1) and a local maximum at (3, 3).

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

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.

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.

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.

Textbook Question

109. Suppose the derivative of the function y = f(x) is

y'=(x-1)^2(x-2).

At what points, if any, does the graph of f have a local minimum, local maximum, or

point of inflection? (Hint: Draw the sign pattern for y'.)

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

Key Concepts

Differentiable Function

Local Minima

Graph Sketching

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Theory and Examples

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