Explanation Given Information: The provided equation is: f ( x ) = 2 x 2 − 24 x + 4 Formula used: First Derivative Test: Suppose that c is a critical point of the continuous function f and its derivative is defined for x close to, and on both sides of x = c . Then, determine the sign of the derivative to the left and the right of x = c . 1. If f ' ( x ) is positive to the left of x = c and negative to the right, then f has a maximum at x = c . 2. If f ' ( x ) is negative to the left of x = c and positive to the right, then f has a minimum at x = c . 3. If f ' ( x ) has the same sign on both sides of x = c , then f has neither a maximum nor a minimum at x = c . Stationary Points: Points in the interior of the domain where the derivative is zero. To locate stationary points set f ' ( x ) = 0 and solve for x . Singular Points: Points in the interior of the domain where the derivative is not defined. To locate singular points, find the interior points x where f ' ( x ) is not defined but f ( x ) is defined. End Points: Closed intervals contain end points but open intervals do not have end points. Calculation: Consider the provided equation: f ( x ) = 2 x 2 − 24 x + 4 Differentiating both sides of the equation with respect to x to get: f ' ( x ) = 4 x ( x + 4 ) − ( 2 x 2 − 24 ) ( x + 4 ) 2 f ' ( x ) = 2 x 2 + 16 x + 24 ( x + 4 ) 2 Locating singular points to get: f ' ( x ) = 0 2 x 2 + 16 x + 24 ( x + 4 ) 2 = 0 Multiplying both sides by ( x + 4 ) 2 to get: 2 x 2 + 16 x + 24 ( x + 4 ) 2 ⋅ ( x + 4 ) 2 = 0 2 x 2 + 16 x + 24 = 0 Factorizing the above equation to get: 2 x 2 + 12 x + 4 x + 24 = 0 ( 2 x + 4 ) ( x + 6 ) = 0 Thus, 2 x + 4 = 0 Subtracting 4 from both sides of the equation to get: 2 x + 4 − 4 = 0 − 4 2 x = − 4 Dividing both sides of the equation by 2 to get: 2 x 2 = − 4 2 x = − 2 Or x + 6 = 0 Subtracting 6 from both sides of the equation to get: x + 6 − 6 = 0 − 6 x = − 6 The equation f ' ( x ) = 2 x 2 + 16 x + 24 ( x + 4 ) 2 is always defined so there are no singular points. Locating stationary points to get: The equation has a stationary point at x = − 2 and x = − 6 . Locating End points: There are no endpoints. The values of x are recorded in a table, together with the corresponding y − c o o r d i n a t e s (values of f )

7th Edition

ISBN: 9781337291248

Author: Waner, Stefan.

Publisher: Cengage Learning,

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 5.1, Problem 41E

To determine

To calculate: The exact location of the relative and absolute extrema of the function f(x)=2x2−24x+4.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.1, Problem 40E

Chapter 5.1, Problem 42E

Chapter 5 Solutions

Applied Calculus

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The exact location of the relative and absolute extrema of the function f ( x ) = 2 x 2 − 24 x + 4 .

Solution Summary: The author explains how to calculate the relative and absolute extrema of the function f(x).

Concept explainers

Videos

To calculate: The exact location of the relative and absolute extrema of the function f(x)=2x2−24x+4.

Want to see the full answer?

Chapter 5 Solutions