Explanation Given information: f ( x ) = x 4 − 2 a x 2 + b Definition used: A point at which the graph of a function f changes concavity is called an inflection point. At the inflection points, f ' ' is zero or undefined. For any function f , a point p in the domain of f where f ' ( p ) = 0 or f ' ( p ) is undefined is called a critical point Of the function. Calculation: f ( x ) = x 4 − 2 a x 2 + b (Given) Differentiating with respect to ‘x’, we get f ' ( x ) = 4 x 3 − 4 a x = 4 x ( x 2 − a ) f ' ' ( x ) = 12 x 2 − 4 a f ' ( x ) = To determine (b) To find: The values of the parameters a and b if f has the critical point at ( 2 , 5 ) . To determine (c) To find: The inflection point if the critical is ( 2 , 5 ) .

6th Edition

ISBN: 9781119267874

Author: Hughes-Hallett

Publisher: WILEY

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 4.2, Problem 23P

To determine

(a)

To find:

All critical points and all inflection points of the function f(x)=x4−2ax2+b where a and b are constant.

To determine

(b)

To find:

The values of the parameters a and b if f has the critical point at (2,5).

To determine

(c)

To find:

The inflection point if the critical is (2,5).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.2, Problem 22P

Chapter 4.2, Problem 24P

Students have asked these similar questions

Find the general solution to the differential equation

charity savings Budget for May travel food Peter earned $700 during May. The graph shows how the money was used. What fraction was clothes? O Search Submit clothes leisure

Exercise 11.3 A slope field is given for the equation y' = 4y+4. (a) Sketch the particular solution that corresponds to y(0) = −2 (b) Find the constant solution (c) For what initial conditions y(0) is the solution increasing? (d) For what initial conditions y(0) is the solution decreasing? (e) Verify these results using only the differential equation y' = 4y+4.

Chapter 4 Solutions

APPLIED CALCULUS-WILEYPLUS

Ch. 4.1 - Prob. 1P Ch. 4.1 - Prob. 2P Ch. 4.1 - Prob. 3P Ch. 4.1 - Prob. 4P Ch. 4.1 - Prob. 5P Ch. 4.1 - Prob. 6P Ch. 4.1 - Prob. 7P Ch. 4.1 - Prob. 8P Ch. 4.1 - Prob. 9P Ch. 4.1 - Prob. 10P

Ch. 4.1 - Prob. 11P Ch. 4.1 - Prob. 12P Ch. 4.1 - Prob. 13P Ch. 4.1 - Prob. 14P Ch. 4.1 - Prob. 15P Ch. 4.1 - Prob. 16P Ch. 4.1 - Prob. 17P Ch. 4.1 - Prob. 18P Ch. 4.1 - Prob. 19P Ch. 4.1 - Prob. 20P Ch. 4.1 - Prob. 21P Ch. 4.1 - Prob. 22P Ch. 4.1 - Prob. 23P Ch. 4.1 - Prob. 24P Ch. 4.1 - Prob. 25P Ch. 4.1 - Prob. 26P Ch. 4.1 - Prob. 27P Ch. 4.1 - Prob. 28P Ch. 4.1 - Prob. 29P Ch. 4.1 - Prob. 30P Ch. 4.1 - Prob. 31P Ch. 4.1 - Prob. 32P Ch. 4.1 - Prob. 33P Ch. 4.1 - Prob. 34P Ch. 4.1 - Prob. 35P Ch. 4.1 - Prob. 36P Ch. 4.1 - Prob. 37P Ch. 4.1 - Prob. 38P Ch. 4.1 - Prob. 39P Ch. 4.1 - Prob. 40P Ch. 4.1 - Prob. 41P Ch. 4.1 - Prob. 42P Ch. 4.1 - Prob. 43P Ch. 4.1 - Prob. 44P Ch. 4.1 - Prob. 45P Ch. 4.1 - Prob. 46P Ch. 4.1 - Prob. 47P Ch. 4.1 - Prob. 48P Ch. 4.1 - Prob. 49P Ch. 4.1 - Prob. 50P Ch. 4.1 - Prob. 51P Ch. 4.1 - Prob. 52P Ch. 4.1 - Prob. 53P Ch. 4.1 - Prob. 54P Ch. 4.1 - Prob. 55P Ch. 4.1 - Prob. 56P Ch. 4.1 - Prob. 57P Ch. 4.1 - Prob. 58P Ch. 4.2 - Prob. 1P Ch. 4.2 - Prob. 2P Ch. 4.2 - Prob. 3P Ch. 4.2 - Prob. 4P Ch. 4.2 - Prob. 5P Ch. 4.2 - Prob. 6P Ch. 4.2 - Prob. 7P Ch. 4.2 - Prob. 8P Ch. 4.2 - Prob. 9P Ch. 4.2 - Prob. 10P Ch. 4.2 - Prob. 11P Ch. 4.2 - Prob. 12P Ch. 4.2 - Prob. 13P Ch. 4.2 - Prob. 14P Ch. 4.2 - Prob. 15P Ch. 4.2 - Prob. 16P Ch. 4.2 - Prob. 17P Ch. 4.2 - Prob. 18P Ch. 4.2 - Prob. 19P Ch. 4.2 - Prob. 20P Ch. 4.2 - Prob. 21P Ch. 4.2 - Prob. 22P Ch. 4.2 - Prob. 23P Ch. 4.2 - Prob. 24P Ch. 4.2 - Prob. 25P Ch. 4.2 - Prob. 26P Ch. 4.2 - Prob. 27P Ch. 4.2 - Prob. 28P Ch. 4.2 - Prob. 29P Ch. 4.2 - Prob. 30P Ch. 4.2 - Prob. 31P Ch. 4.2 - Prob. 32P Ch. 4.2 - Prob. 33P Ch. 4.2 - Prob. 34P Ch. 4.2 - Prob. 35P Ch. 4.2 - Prob. 36P Ch. 4.2 - Prob. 37P Ch. 4.2 - Prob. 38P Ch. 4.2 - Prob. 39P Ch. 4.2 - Prob. 40P Ch. 4.2 - Prob. 41P Ch. 4.3 - Prob. 1P Ch. 4.3 - Prob. 2P Ch. 4.3 - Prob. 3P Ch. 4.3 - Prob. 4P Ch. 4.3 - Prob. 5P Ch. 4.3 - Prob. 6P Ch. 4.3 - Prob. 7P Ch. 4.3 - Prob. 8P Ch. 4.3 - Prob. 9P Ch. 4.3 - Prob. 10P Ch. 4.3 - Prob. 11P Ch. 4.3 - Prob. 12P Ch. 4.3 - Prob. 13P Ch. 4.3 - Prob. 14P Ch. 4.3 - Prob. 15P Ch. 4.3 - Prob. 16P Ch. 4.3 - Prob. 17P Ch. 4.3 - Prob. 18P Ch. 4.3 - Prob. 19P Ch. 4.3 - Prob. 20P Ch. 4.3 - Prob. 21P Ch. 4.3 - Prob. 22P Ch. 4.3 - Prob. 23P Ch. 4.3 - Prob. 24P Ch. 4.3 - Prob. 25P Ch. 4.3 - Prob. 26P Ch. 4.3 - Prob. 27P Ch. 4.3 - Prob. 28P Ch. 4.3 - Prob. 29P Ch. 4.3 - Prob. 30P Ch. 4.3 - Prob. 31P Ch. 4.3 - Prob. 32P Ch. 4.3 - Prob. 33P Ch. 4.3 - Prob. 34P Ch. 4.3 - Prob. 35P Ch. 4.3 - Prob. 36P Ch. 4.3 - Prob. 37P Ch. 4.3 - Prob. 38P Ch. 4.3 - Prob. 39P Ch. 4.3 - Prob. 40P Ch. 4.3 - Prob. 41P Ch. 4.3 - Prob. 42P Ch. 4.3 - Prob. 43P Ch. 4.3 - Prob. 44P Ch. 4.3 - Prob. 45P Ch. 4.3 - Prob. 46P Ch. 4.3 - Prob. 47P Ch. 4.3 - Prob. 48P Ch. 4.3 - Prob. 49P Ch. 4.3 - Prob. 50P Ch. 4.3 - Prob. 51P Ch. 4.3 - Prob. 52P Ch. 4.3 - Prob. 53P Ch. 4.3 - Prob. 54P Ch. 4.3 - Prob. 55P Ch. 4.3 - Prob. 56P Ch. 4.3 - Prob. 57P Ch. 4.3 - Prob. 58P Ch. 4.3 - Prob. 59P Ch. 4.3 - Prob. 60P Ch. 4.4 - Prob. 1P Ch. 4.4 - Prob. 2P Ch. 4.4 - Prob. 3P Ch. 4.4 - Prob. 4P Ch. 4.4 - Prob. 5P Ch. 4.4 - Prob. 6P Ch. 4.4 - Prob. 7P Ch. 4.4 - Prob. 8P Ch. 4.4 - Prob. 9P Ch. 4.4 - Prob. 10P Ch. 4.4 - Prob. 11P Ch. 4.4 - Prob. 12P Ch. 4.4 - Prob. 13P Ch. 4.4 - Prob. 14P Ch. 4.4 - Prob. 15P Ch. 4.4 - Prob. 16P Ch. 4.4 - Prob. 17P Ch. 4.4 - Prob. 18P Ch. 4.4 - Prob. 19P Ch. 4.4 - Prob. 20P Ch. 4.4 - Prob. 21P Ch. 4.4 - Prob. 22P Ch. 4.4 - Prob. 23P Ch. 4.4 - Prob. 24P Ch. 4.4 - Prob. 25P Ch. 4.4 - Prob. 26P Ch. 4.4 - Prob. 27P Ch. 4.4 - Prob. 28P Ch. 4.4 - Prob. 29P Ch. 4.4 - Prob. 30P Ch. 4.4 - Prob. 31P Ch. 4.4 - Prob. 32P Ch. 4.4 - Prob. 33P Ch. 4.4 - Prob. 34P Ch. 4.4 - Prob. 35P Ch. 4.5 - Prob. 1P Ch. 4.5 - Prob. 2P Ch. 4.5 - Prob. 3P Ch. 4.5 - Prob. 4P Ch. 4.5 - Prob. 5P Ch. 4.5 - Prob. 6P Ch. 4.5 - Prob. 7P Ch. 4.5 - Prob. 8P Ch. 4.5 - Prob. 9P Ch. 4.5 - Prob. 10P Ch. 4.5 - Prob. 11P Ch. 4.5 - Prob. 12P Ch. 4.5 - Prob. 13P Ch. 4.5 - Prob. 14P Ch. 4.5 - Prob. 15P Ch. 4.5 - Prob. 16P Ch. 4.5 - Prob. 17P Ch. 4.6 - Prob. 1P Ch. 4.6 - Prob. 2P Ch. 4.6 - Prob. 3P Ch. 4.6 - Prob. 4P Ch. 4.6 - Prob. 5P Ch. 4.6 - Prob. 6P Ch. 4.6 - Prob. 7P Ch. 4.6 - Prob. 8P Ch. 4.6 - Prob. 9P Ch. 4.6 - Prob. 10P Ch. 4.6 - Prob. 11P Ch. 4.6 - Prob. 12P Ch. 4.6 - Prob. 13P Ch. 4.6 - Prob. 14P Ch. 4.6 - Prob. 15P Ch. 4.6 - Prob. 16P Ch. 4.6 - Prob. 17P Ch. 4.6 - Prob. 18P Ch. 4.6 - Prob. 19P Ch. 4.6 - Prob. 20P Ch. 4.6 - Prob. 21P Ch. 4.6 - Prob. 22P Ch. 4.6 - Prob. 23P Ch. 4.6 - Prob. 24P Ch. 4.6 - Prob. 25P Ch. 4.6 - Prob. 26P Ch. 4.6 - Prob. 27P Ch. 4.6 - Prob. 28P Ch. 4.6 - Prob. 29P Ch. 4.6 - Prob. 30P Ch. 4.6 - Prob. 31P Ch. 4.7 - Prob. 1P Ch. 4.7 - Prob. 2P Ch. 4.7 - Prob. 3P Ch. 4.7 - Prob. 4P Ch. 4.7 - Prob. 5P Ch. 4.7 - Prob. 6P Ch. 4.7 - Prob. 7P Ch. 4.7 - Prob. 8P Ch. 4.7 - Prob. 9P Ch. 4.7 - Prob. 10P Ch. 4.7 - Prob. 11P Ch. 4.7 - Prob. 12P Ch. 4.7 - Prob. 13P Ch. 4.7 - Prob. 14P Ch. 4.7 - Prob. 15P Ch. 4.7 - Prob. 16P Ch. 4.7 - Prob. 17P Ch. 4.7 - Prob. 18P Ch. 4.7 - Prob. 19P Ch. 4.7 - Prob. 20P Ch. 4.7 - Prob. 21P Ch. 4.7 - Prob. 22P Ch. 4.7 - Prob. 23P Ch. 4.7 - Prob. 24P Ch. 4.8 - Prob. 1P Ch. 4.8 - Prob. 2P Ch. 4.8 - Prob. 3P Ch. 4.8 - Prob. 4P Ch. 4.8 - Prob. 5P Ch. 4.8 - Prob. 6P Ch. 4.8 - Prob. 7P Ch. 4.8 - Prob. 8P Ch. 4.8 - Prob. 9P Ch. 4.8 - Prob. 10P Ch. 4.8 - Prob. 11P Ch. 4.8 - Prob. 12P Ch. 4.8 - Prob. 13P Ch. 4 - Prob. 1SYU Ch. 4 - Prob. 2SYU Ch. 4 - Prob. 3SYU Ch. 4 - Prob. 4SYU Ch. 4 - Prob. 5SYU Ch. 4 - Prob. 6SYU Ch. 4 - Prob. 7SYU Ch. 4 - Prob. 8SYU Ch. 4 - Prob. 9SYU Ch. 4 - Prob. 10SYU Ch. 4 - Prob. 11SYU Ch. 4 - Prob. 12SYU Ch. 4 - Prob. 13SYU Ch. 4 - Prob. 14SYU Ch. 4 - Prob. 15SYU Ch. 4 - Prob. 16SYU Ch. 4 - Prob. 17SYU Ch. 4 - Prob. 18SYU Ch. 4 - Prob. 19SYU Ch. 4 - Prob. 20SYU Ch. 4 - Prob. 21SYU Ch. 4 - Prob. 22SYU Ch. 4 - Prob. 23SYU Ch. 4 - Prob. 24SYU Ch. 4 - Prob. 25SYU Ch. 4 - Prob. 26SYU Ch. 4 - Prob. 27SYU Ch. 4 - Prob. 28SYU Ch. 4 - Prob. 29SYU Ch. 4 - Prob. 30SYU Ch. 4 - Prob. 31SYU Ch. 4 - Prob. 32SYU Ch. 4 - Prob. 33SYU Ch. 4 - Prob. 34SYU Ch. 4 - Prob. 35SYU Ch. 4 - Prob. 36SYU Ch. 4 - Prob. 37SYU Ch. 4 - Prob. 38SYU Ch. 4 - Prob. 39SYU Ch. 4 - Prob. 40SYU Ch. 4 - Prob. 41SYU Ch. 4 - Prob. 42SYU Ch. 4 - Prob. 43SYU Ch. 4 - Prob. 44SYU Ch. 4 - Prob. 45SYU Ch. 4 - Prob. 46SYU Ch. 4 - Prob. 47SYU Ch. 4 - Prob. 48SYU Ch. 4 - Prob. 49SYU Ch. 4 - Prob. 50SYU Ch. 4 - Prob. 51SYU Ch. 4 - Prob. 52SYU Ch. 4 - Prob. 53SYU Ch. 4 - Prob. 54SYU Ch. 4 - Prob. 55SYU Ch. 4 - Prob. 56SYU Ch. 4 - Prob. 57SYU Ch. 4 - Prob. 58SYU Ch. 4 - Prob. 59SYU Ch. 4 - Prob. 60SYU Ch. 4 - Prob. 61SYU Ch. 4 - Prob. 62SYU Ch. 4 - Prob. 63SYU Ch. 4 - Prob. 64SYU Ch. 4 - Prob. 65SYU Ch. 4 - Prob. 66SYU Ch. 4 - Prob. 67SYU Ch. 4 - Prob. 68SYU Ch. 4 - Prob. 69SYU Ch. 4 - Prob. 70SYU Ch. 4 - Prob. 71SYU Ch. 4 - Prob. 72SYU Ch. 4 - Prob. 73SYU Ch. 4 - Prob. 74SYU Ch. 4 - Prob. 75SYU Ch. 4 - Prob. 76SYU Ch. 4 - Prob. 77SYU Ch. 4 - Prob. 78SYU Ch. 4 - Prob. 79SYU Ch. 4 - Prob. 80SYU

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.

(a) To find: All critical points and all inflection points of the function f ( x ) = x 4 − 2 a x 2 + b where a and b are constant.

Concept explainers

Videos

(a) To find: All critical points and all inflection points of the function f(x)=x4−2ax2+b where a and b are constant.

(b) To find: The values of the parameters a and b if f has the critical point at (2,5).

(c) To find: The inflection point if the critical is (2,5).

Want to see the full answer?

Chapter 4 Solutions

(a)

To find:

All critical points and all inflection points of the function f(x)=x4−2ax2+b where a and b are constant.

(b)

To find:

The values of the parameters a and b if f has the critical point at (2,5).

(c)

To find:

The inflection point if the critical is (2,5).