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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.1.41
Chapter 4, Problem 4.1.41

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


ƒ(t) = 1/5 t⁵ - a⁴t

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To find the critical points of the function ƒ(t) = 1/5 t⁵ - a⁴t, we first need to find its derivative ƒ'(t). The derivative will help us identify where the slope of the tangent to the curve is zero or undefined.
Differentiate the function ƒ(t) = 1/5 t⁵ - a⁴t with respect to t. Using the power rule, the derivative of 1/5 t⁵ is (1/5) * 5 * t⁴ = t⁴. The derivative of -a⁴t is -a⁴. Therefore, ƒ'(t) = t⁴ - a⁴.
Set the derivative ƒ'(t) equal to zero to find the critical points: t⁴ - a⁴ = 0. This equation will help us find the values of t where the function has critical points.
Solve the equation t⁴ - a⁴ = 0 for t. This can be rewritten as t⁴ = a⁴. Taking the fourth root of both sides, we find t = ±a.
The critical points of the function are t = a and t = -a. These are the values of t where the derivative is zero, indicating potential maxima, minima, or points of inflection.

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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 the variable that satisfy these conditions.
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Derivative

The derivative of a function measures the rate at which the function's value changes with respect to changes in its input. It is a fundamental concept in calculus, providing insights into the function's behavior, such as increasing or decreasing intervals. For the given function, the derivative will be calculated to find the critical points.
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Derivatives

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function given, ƒ(t) = 1/5 t⁵ - a⁴t, is a polynomial of degree five. Understanding the properties of polynomial functions, such as their continuity and differentiability, is crucial for analyzing their critical points.
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Introduction to Polynomial Functions
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