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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2:43 minutes

Problem 4.1.29

Textbook Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x³ -4a²x

Verified step by step guidance

To find the critical points of the function \( f(x) = x^3 - 4a^2x \), we first need to find its derivative. The derivative, \( f'(x) \), will help us identify where the slope of the tangent to the curve is zero or undefined.

Differentiate the function \( f(x) = x^3 - 4a^2x \) with respect to \( x \). Using the power rule, the derivative is \( f'(x) = 3x^2 - 4a^2 \).

Set the derivative \( f'(x) = 3x^2 - 4a^2 \) equal to zero to find the critical points. This gives the equation \( 3x^2 - 4a^2 = 0 \).

Solve the equation \( 3x^2 - 4a^2 = 0 \) for \( x \). This can be done by adding \( 4a^2 \) to both sides and then dividing by 3, resulting in \( x^2 = \frac{4a^2}{3} \).

Take the square root of both sides to solve for \( x \). Remember to consider both the positive and negative roots, so \( x = \pm \sqrt{\frac{4a^2}{3}} \). These are the critical points of the function.

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Key Concepts

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

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local maxima, minima, and points of inflection. To find critical points, one typically takes the derivative of the function and solves for the values of x that satisfy the condition.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve at any given point. For polynomial functions, the derivative can be calculated using power rules, which simplify the process of finding critical points.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function given, ƒ(x) = x³ - 4a²x, is a cubic polynomial. Understanding the behavior of polynomial functions, including their derivatives, is crucial for analyzing their critical points and overall shape.

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{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.

b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.