Determine and classify all critical points , Mathematics

Assignment Help:

Determine and classify all the critical points of the given function.  Described the intervals where function is increasing & decreasing.

423_Classifying critical points.png

Solution: Firstly we'll require the derivative therefore we can get our hands on the critical points.  Note that we'll do some simplification on the derivative to help us find out the critical points.

g ′ (t ) = (t 2 - 4)(1/3)  +(2/3) t 2 (t 2 - 4)- 3

=(t 2 - 4)3  +     2t2 / 3(t 2 - 4)(2/3)

=3 (t 2 - 4) + 2t 2 /3(t 2 - 4)(2/3)

 =5t 2  -12 /3 (t 2 - 4)3

Thus, it looks like we'll have four critical points here. They are following,

t =± 2                                      here, the derivative doesn't exist.

t = ±√(12/5) = ±1.549             here, the derivative is zero here.

2026_critical point.png

Determining the intervals of increasing & decreasing will also give the classification of the critical points thus let's get those first.  Given is a number line along with the critical points graphed & test points.

Thus, we've got the given intervals of increasing & decreasing.

Increase : - ∞ < x - 2, - 2 < x < -√(12/5) , √(12/5) < x < 2,  &  2 < x < ∞

Decrease :  -√(12/5)

From this it looks like t = -2 & t = 2 are neither relative minimum or relative maximums as the function is increasing on both side of them.  Alternatively, t < -12   is a relative maximum and t =12 is a relative minimum.

Here is the graph of the function.

420_cp1.png


Related Discussions:- Determine and classify all critical points

Determine if the three vectors lie in similar plane or not, Determine if th...

Determine if the three vectors a → = (1, 4, -7), b → = (2, -1, 4) and c → = (0, -9, 18) lie in similar plane or not. Solution Thus, as we noted prior to this example al

Do all our activities involve mathematics?, Do All Our Activities Involve M...

Do All Our Activities Involve Mathematics? :  The answer to this is 'yes' and 'no'. For those who look for mathematics and know where to look for it, it is 'yes'. For those who do

Interpretation of r – problems in interpreting r values, Interpretation of ...

Interpretation of r - Problems in interpreting r values A high value of r as +0.9 or - 0.9 only shows a strong association among the two variables but doesn't imply that th

Sequences - calculus, Sequences Let us start off this section along wi...

Sequences Let us start off this section along with a discussion of just what a sequence is. A sequence is nothing much more than a list of numbers written in a particular orde

Trigonometry, explain the formular for finding trigonometry

explain the formular for finding trigonometry

How long will it take her to save $350, Each week Jaime saves $25. How long...

Each week Jaime saves $25. How long will it take her to save $350? Divide $350 by $25; 350 ÷ 25 = 14 weeks.

Create a table with the number of components of each size, Look on the web ...

Look on the web for a data base that can be converted to an undirected graph.  For  example, in Science there is a data base of proteins and their interactions.  Each protein can b

Differential equations and group methods, solve the differential equation ...

solve the differential equation dy/dx=f(y)x^n+g(y)x^m by finding a one-parameter group leaving it invariant

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd