Find intervals while function is increasing or decreasing, Mathematics

Assignment Help:

Find out all intervals where the given function is increasing or decreasing.

f ( x ) = - x5 + 5/2 x4 + 40/3 x3 + 5

Solution

To find out if the function is increasing or decreasing we will required the derivative.

f ′ ( x ) = -5x4 + 10x3 + 40x2

= -5x2 ( x2 - 2x - 8)

= -5x2 ( x - 4) ( x + 2)

Note as well that while we factored the derivative first we factored a "-1" out to make the rest of the factoring a little bit easier.

From the factored form of the derivative we illustrates that we have three critical points :x = -2 ,x = 0 , and x = 4 .  We'll require these in a bit.

Now we have to determine where the derivative is +ve and where it's -ve..  As the derivative is a polynomial this is continuous and thus we know that the only way for it to change signs is to primary go through zero.

In other terms, the only place that the derivative might change signs is at the critical points of the function.  Now we've got another utilizes for critical points.  Thus, we'll build a number line, graph the critical points & pick test points through each region to illustrate if the derivative is positive or negative in each of region.

Following is the number line & the test points for the derivative.

316_function 1.png

Ensure that you test your points in the derivative. Recall that we know that the derivative will be the similar sign in each of region. The only place that the derivative can alter signs is at the critical points & we've marked the only critical points onto the number line.

Thus, it looks we've got the given intervals of increase & decrease.

Increase : - 2 < x <0 and 0 < x < 4

Decrease : - ∞ < x < -2 and 4 < x < ∞


Related Discussions:- Find intervals while function is increasing or decreasing

Find the distance of the bird from the girl, A boy standing on a horizontal...

A boy standing on a horizontal plane finds a bird flying at a distance of 100m from him at an elevation of 300. A girl standing on the roof of 20 meter high building finds the angl

Illustration of integration by parts - integration technique, Example of In...

Example of Integration by Parts - Integration techniques Some problems could need us to do integration by parts many times and there is a short hand technique that will permit

Determine principal strains and direction , A 100 by 150 mm rectangular pla...

A 100 by 150 mm rectangular plate is deformed as shown in the following figure. All dimensions shown in the figure are in millimeters.  Determine at point Q: (a) the strain compone

Four is added to the quantity two minus the sum of negative, Four is added ...

Four is added to the quantity two minus the sum of negative seven and six. This answer is then multiplied through three. What is the result? This problem translates to the expr

Sqrt n- sqrt 8836, How many integers satisfy (sqrt n- sqrt 8836)^2 Solutio...

How many integers satisfy (sqrt n- sqrt 8836)^2 Solution) sqrt 8836 = 94 , let sqrt n=x the equation becomes... (x-94)^2 (x-94)^2 - 1 (x-95)(x-93) hence  93 8649  the number o

Algorithm for division, ALGORITHM FOR DIVISION : If you ask a 10 or 1 1-ye...

ALGORITHM FOR DIVISION : If you ask a 10 or 1 1-year-old child to solve, say, 81 + 9, the chances are that she will correctly do it. But if you ask her to solve, say 72 + 3, t

Determine the area of the rectangle, Stuckeyburg is a very small town in ru...

Stuckeyburg is a very small town in rural America. Use the map to approximate the area of the town. a. 40 miles 2 b. 104 miles 2 c. 93.5 miles 2 d. 92 miles 2

Find a general solution to the differential equation, Example: Find a gene...

Example: Find a general solution to the subsequent differential equation. 2 y′′ + 18 y + 6 tan (3t) Solution First, as the formula for variation of parameters needs coe

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