The function f given by f(x)=3x5−4x3−3x has a relative maximum at x?

Answer:

We have the function:

f(x) = 3*x^5 - 4*x^3 - 3*x

To find the maximum/minimums, we first need to look at the roots of the first derivative of f(x), this is:

f´(x) = 5*3*x^4 - 3*4*x^2 - 3

f´(x) = 15*x^4 - 12*x^2 - 3

We can do the change of variable:

u = x^2

Then we can rewrite:

f'(u) = 15*u^2 - 12*u - 3

This is a quadratic equation, the solutions will be obtained by the Bhaskara´s equation:

So the solutions are:

[tex]u = \frac{+12 +- \sqrt{(-12)^2 - 4*(-3)*15} }{2*15} = \frac{12 +- 18}{30}[/tex]

Then the two solutions are:

u = (12 + 18)/30 = 1

this leads to:

x^2 = 1, then we can have x = ± 1

the other solution is:

u = (12 - 18)/30 = -6/30 = -(1/5)

and x^2 = -(1/5), then:

x = ±√(1/5)*i

So those are the possible values for the minimum/maximums.

Now to see it, we need to look at the second derivative ad see iff:

f´´(x) > 0 then x is a minimum

f''(x) < 0, then x is a maximum.

We have that:

f´´(x) = 4*15*x^3 - 2*12*x

f´´(x) = 60*x^3 - 24*x

Notice that the two complex roots will lead to complex values of f(x), so these can be ignored here, then we have to try only with x = ± 1.

f´´(1) = 60*1^3 - 24*1 = 36

then x = 1 is a minimum.

f´´(-1) = 60*(-1)^3 -24*-1 = -36

then x = -1 is a maximum.

Then we can conclude that the relative maximum is at x = -1

The function f given by [tex]f(x) = 3x^5-4x^3-3x[/tex] has a relative maximum at

x = -1

The given function is:

[tex]f(x) = 3x^5-4x^3-3x[/tex]

Find the derivative of the function:

[tex]f'(x)=15x^4-12x^2-3[/tex]

Find the critical points by equation f'(x) to zero

[tex]15x^4-12x^2-3=0\\\\5x^4-4x^2-1=0[/tex]

By solving the equation above, the critical points are:

x = -1,  x =  1

Find the second derivative of f(x)

[tex]f''(x)=60x^3-24x[/tex]

[tex]f''(-1)=60(-1)^3-24(-1)\\\\f''(-1)=-36[/tex]

Since f''(-1) < 0, x = -1 is a maximum point

[tex]f''(1)=60(1)^3-24(1)\\\\f''(1)=36[/tex]

Since f''(1) > 0, x = 1 is a minimum point

Therefore, the function [tex]f(x) = 3x^5-4x^3-3x[/tex] has a relative maximum at x = -1

Learn more here: https://brainly.com/question/19635205

Help please!!!

Mr. Hayes has 111 straw bundles to use for bedding in his horse stalls. what is the greatest number of stalls he can fill if each stall holds exactly 4 bundles of straw?​

A card is drawn from a​ well-shuffled deck of 52 cards. Find the probability that the card is a red card or a 6?

Johannes needs to buy replacement team uniforms and is looking at different prices he has a budget of 300 he needs 18 pairs of plants and 12 shirts let x represent

Nikki is solving the equation below. Which value of x makes the equation true ?

six friends share a 15 once bag of nuts equally between witch too whole numbers will each get

I need help anyone know what this is?

a line passes through tge points (1,-3) and (4,6) what is the zero of this line ?

What is the common difference between the terms in this arithmetic sequence?

5, 11, 17, 23, 29, . . .

Helppppppppppppppp...

Jacinta buys 4 pounds of turkey and 2 pounds of ham. She pays a total of $30, and the turkey costs $1.50 less per pound than the ham. What would be the combined cost of 1 pound of turkey and 1 pound of ham?
A. $7.50 B. $9.50
C. $10.50 D. $15.00

Danielle created a painting using only similar hexagons.

Which statement about the hexagons must be true?

The hexagons are all the same size and shape.

The hexagons have corresponding sides that are proportional.

The hexagons have corresponding angles that are not congruent.

The hexagons are different shapes but the same size.

You have discovered another alien. Find the correct rescue path before answering this question. Which statements are true about the equation of the line used to save this alien? The y-intercept is not visible on the graph. The slope does not exist for this line. The y-intercept is positive. The line crosses the x-axis at –3.

please help me asap​

12x-20y=180 slope-intercept form

Need it ASAP will give brainliest

A researcher compared the heights and shoe sizes for 50 men, selected at random. The equation shown describes a line of best fit for the data, where x is the shoe size and y is the height, in inches, Based on the equation, what is the approximate shoe size for a man having a height of 60 inches? On O n. 108 O c. 12 D. 14​

Please don’t just comment for points

helppppppppppp pleaseeeeeee ​

I need help with this

Find the value of c that completes the square.
m^2 + 32m + c

Karen is planting a tree. She wants to use three guide wires to stabilize the tree. If Karen places the guide wires 8 feet up
the trunk and 6 feet from the base, how much total wire will she need for both guide wires?

21 pounds into ounces

Please help with this. THank you

You are to take a multiple-choice exam consisting of 100 questions with 2 possible responses to each question. Let x represent the number of correct responses on the test. A) What is your expected score on the exam?B) Compute the variance and standard deviation of x.

help is needed :) similar or not similar

Meg found another treasure between −13
4
feet and
-21
2
feet. Where was that treasure?

Negative 1 and one-half
Negative 2 and three-fourths
Negative 1 and one-fourth
Negative 2 and one-fourth

What is the least common multiple (LCM) of 3 and 5?
O 30
O 2
O 15
O 8
pls help ​

Find the midpoint of the line segment that has endpoints at (-9, -6) and (-1,-8).

PLEASE HELPP Which equation is parallel to y = –3 x + 4 and passes through the point (–3 , 4)? y = –3 x – 5 y = 13 x + 3 y = –3 x + 15 y = 13 x + 5

State if the two triangles are congruent. If they are, state the triangle congruence theorem that proves it