Explanation Given Information: The provided function is f ( x ) = x 4 − 160 x 3 + 7200 x 2 − 40 , 000 . Formula Used: To obtain the relative maxima and minima of a function, 1. Find the first derivative of the function. 2. Set the derivative equal to 0 to obtain the critical points. 3. Evaluate f ′ ( x ) at values of x to the left and one to the right of each critical point to develop a sign diagram. (a) If f ′ ( x ) > 0 to the left and f ′ ( x ) < 0 to the right of the critical value, the critical point is a relative maximum point. (b) If f ′ ( x ) < 0 to the left and f ′ ( x ) > 0 to the right of the critical value, the critical point is a relative minimum point. Calculation: Consider the provided function f ( x ) = x 4 − 160 x 3 + 7200 x 2 − 40 , 000 , Use the simple power rule to differentiate, d y d x = 4 x 3 − 480 x 2 − 14400 x = 4 x ( x 2 − 120 x + 3600 ) = 4 x ( x − 60 ) ( x − 60 ) Equate the above derivative to 0, y ′ = d y d x = 4 x ( x − 60 ) ( x − 60 ) = 0 Evaluate the values of x from the equation: 4 x ( x − 60 ) ( x − 60 ) = 0 x = 0 , 60 Hence, the value(s) of x are x = 0 , 60 . Evaluate the values of the original functions with the critical values: Substitute 0 for x in f ( x ) = x 4 − 160 x 3 + 7200 x 2 − 40 , 000 . y = ( 0 ) 4 − 160 ( 0 ) 3 + 7200 ( 0 ) 2 − 40 , 000 = 0 − 0 + 0 − 40 , 000 = − 40 , 000 Hence, ( 0 , − 40000 ) is a critical point. Substitute 0 for x in f ( x ) = x 4 − 160 x 3 + 7200 x 2 − 40 , 000 . y = ( 60 ) 4 − 160 ( 60 ) 3 + 7200 ( 60 ) 2 − 40 , 000 = 12960000 + 34560000 + 25920000 − 40 , 000 = 4280000 Hence, ( 60 , 4280000 ) is a critical point

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 10.1, Problem 39E

To determine

To calculate: The location of critical points and determines viewing window that shows all the features of the graph f(x)=x4−160x3+7200x2−40,000 and, also sketch the complete graph.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.1, Problem 38E

Chapter 10.1, Problem 40E

Students have asked these similar questions

18.9. Let denote the boundary of the rectangle whose vertices are -2-2i, 2-21, 2+i and -2+i in the positive direction. Evaluate each of the following integrals: (a). 之一 dz, (b). dz, (b). COS 2 coz dz, dz (z+1) (d). z 2 +2 dz, (e). (c). (2z+1)zdz, z+ 1 (f). £, · [e² sin = + (2² + 3)²] dz. (2+3)2

We consider the one-period model studied in class as an example. Namely, we assumethat the current stock price is S0 = 10. At time T, the stock has either moved up toSt = 12 (with probability p = 0.6) or down towards St = 8 (with probability 1−p = 0.4).We consider a call option on this stock with maturity T and strike price K = 10. Theinterest rate on the money market is zero.As in class, we assume that you, as a customer, are willing to buy the call option on100 shares of stock for $120. The investor, who sold you the option, can adopt one of thefollowing strategies: Strategy 1: (seen in class) Buy 50 shares of stock and borrow $380. Strategy 2: Buy 55 shares of stock and borrow $430. Strategy 3: Buy 60 shares of stock and borrow $480. Strategy 4: Buy 40 shares of stock and borrow $280.(a) For each of strategies 2-4, describe the value of the investor’s portfolio at time 0,and at time T for each possible movement of the stock.(b) For each of strategies 2-4, does the investor have…

eric pez Xte in z= Therefore, we have (x, y, z)=(3.0000, 83.6.1 Exercise Gauss-Seidel iteration with Start with (x, y, z) = (0, 0, 0). Use the convergent Jacobi i Tol=10 to solve the following systems: 1. 5x-y+z = 10 2x-8y-z=11 -x+y+4z=3 iteration (x Assi 2 Assi 3. 4. x-5y-z=-8 4x-y- z=13 2x - y-6z=-2 4x y + z = 7 4x-8y + z = -21 -2x+ y +5z = 15 4x + y - z=13 2x - y-6z=-2 x-5y- z=-8 realme Shot on realme C30 2025.01.31 22:35 f

Chapter 10 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 10.1 - 1. The x- values of critical points are found...Ch. 10.1 - Prob. 2CP Ch. 10.1 - Prob. 3CP Ch. 10.1 - Prob. 4CP Ch. 10.1 - In Problems 1 and 2, use the indicated points on...Ch. 10.1 - In Problems 1 and 2, use the indicated points on...Ch. 10.1 - 3. Use the graph of in Problem 1 to identify at...Ch. 10.1 - 4. Use the graph of in Problem 2 to identify at...Ch. 10.1 - In Problems 5 and 6, use the sign diagram for to...Ch. 10.1 - In Problems 5 and 6, use the sign diagram for to...

Knowledge Booster

Learn more about

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

The location of critical points and determines viewing window that shows all the features of the graph f ( x ) = x 4 − 160 x 3 + 7200 x 2 − 40 , 000 and, also sketch the complete graph.

Solution Summary: The author calculates the location of critical points and determines viewing window that shows all the features of the graph.

Concept explainers

Videos

To calculate: The location of critical points and determines viewing window that shows all the features of the graph f(x)=x4−160x3+7200x2−40,000 and, also sketch the complete graph.

Want to see the full answer?

Chapter 10 Solutions