Chapter 4: Q1P (page 213)

To find the familiar "second derivative test "for a maximum or minimum point. That is show that $f^{'} (a) = 0$ , then $f^{''} (a) > 0$ implies a minimum point at $x = a$ and $f^{''} (a) < 0$ implies a maximum point at $x = a$ .

Short Answer

Expert verified

For all $x$ near enough to $a$ is given as follows:

$f (x) - f (a) > 0 f (x) > f (a)$

Step by step solution

Given data

The given function is $f^{'} (a) = 0$ , and then $f^{''} (a) > 0$ which implies a minimum point at $x = a$ and $f^{''} (a) < 0$ implies a maximum point at $x = a$ .

Concept of Taylor series

Taylor series representation off about $x = a$ is given as:

$f (x) = f (a) + \frac{f^{'} (a)}{1!} + \frac{f^{''} (a)}{2!} (x - a)^{2}$ …(1)

Calculation to find the maximum point for the derivative test f

Suppose that $f^{'} (a) = 0$ then from (1) obtain:

$f^{''} (a) > 0 2 \frac{f (x) - f (a)}{{(x - a)}^{2}} > 0$

For all $x$ near enough to $a$ is given as follows:

$f (x) - f (a) > 0 f (x) > f (a)$

That is, $f$ increases.

Now, by the first derivative test $f$ has minimum point at $x = a$ :

$f^{''} (a) < 0 \frac{2 f (x) - f (a)}{(x - a)} < 0 f (x) - f (a) < 0$

$f (x) < f (a)$ for all $x$ near enough to $a$ .

That is, $f$ decreases.

Now, by the first derivative test $f$ has maximum point at $x = a$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

To find the familiar "second derivative test "for a maximum or minimum point. That is show that $f^{'} (a) = 0$ , then $f^{''} (a) > 0$ implies a minimum point at $x = a$ and $f^{''} (a) < 0$ implies a maximum point at $x = a$ .

Short Answer

Step by step solution

Given data

Concept of Taylor series

Calculation to find the maximum point for the derivative test f

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Nuclear Physics

Translational Dynamics

Astrophysics

Energy Physics

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

To find the familiar "second derivative test "for a maximum or minimum point. That is show that f'(a)=0, thenf''(a)>0implies a minimum point atx=aandf''(a)<0implies a maximum point at x=a.

Short Answer

Step by step solution

Given data

Concept of Taylor series

Calculation to find the maximum point for the derivative test f

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Nuclear Physics

Translational Dynamics

Astrophysics

Energy Physics

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

To find the familiar "second derivative test "for a maximum or minimum point. That is show that $f^{'} (a) = 0$ , then $f^{''} (a) > 0$ implies a minimum point at $x = a$ and $f^{''} (a) < 0$ implies a maximum point at $x = a$ .