Answer:
[tex]\text{$f(x)$ is increasing for $(-\infty, -\frac{1}{3})$ and $(4, \infty)$} \\ \text{$f(x)$ is decreasing for $(-\frac{1}{3}, 4)$}[/tex]
Step-by-step explanation:
We have the function:
[tex]f(x)=x^3-\frac{11}{2}x^2-4x[/tex]
And we want to determine the intervals for which the function is increasing or decreasing.
We will need to first find the critical points of the function. Note that our original function is continuous across the entire x-axis.
To find the critical points, we need to find the first derivative and then solve for the x. So:
[tex]f^\prime(x)=\frac{d}{dx}[x^3-\frac{11}{2}x^2-4x][/tex]
Differentiate:
[tex]f^\prime(x)=(3x^2)-\frac{11}{2}(2x)-(4) \\ f^\prime(x)=3x^2-11x-4[/tex]
Set the derivative equal to 0 and solve for x:
[tex]0=3x^2-11x-4[/tex]
Factor:
[tex]0=3x^2-12x+x-4 \\ 0=3x(x-4)+(x-4) \\ 0=(3x+1)(x-4)[/tex]
Zero Product Property:
[tex]3x+1=0\text{ or } x-4=0 \\ x=-\frac{1}{3}\text{ or } x=4[/tex]
Therefore, our critical points are -1/3 and 4.
We can thus sketch the following number line:
<————-(-1/3)——————————————(4)—————>
Now, let’s test values for the three intervals: less than -1/3, between -1/3 and 4, and greater than 4.
For less then -1/3, we can use -1. So, substitute -1 for x for our first derivative and see which we get:
[tex]f^\prime(-1)=3(-1)^2-11(-1)-4=3+11-4=10>0[/tex]
The result is positive.
Therefore, for all numbers less than -1/3,f(x) is increasing (since its derivative is positive).
For between -1/3 and 4, we can use 0. Substitute 0 for our derivative:
[tex]f^\prime(0)=3(0)^2-11(0)-4=-4<0[/tex]
The result is negative.
Therefore, for all numbers between -1/3 and 4, f(x) is decreasing (since its derivative is negative).
And finally, for greater than 4, we can use 5:
[tex]f^\prime(5)=3(5)^2-11(5)-4=16>0[/tex]
The result is positive.
Therefore, for all numbers greater than 4, f(x) is increasing (since its derivative is positive.
Therefore:
[tex]\text{$f(x)$ is increasing for $x<-\frac{1}{3}$ and $x>4$}\\ \text{ $f(x)$ is decreasing for $-\frac{1}{3}<x<4$}[/tex]
In interval notation:
[tex]\text{$f(x)$ is increasing for $(-\infty, -\frac{1}{3})$ and $(4, \infty)$} \\ \text{$f(x)$ is decreasing for $(-\frac{1}{3}, 4)$}[/tex]
3) The measure of the angle which is equal to its supplement is
Answer:
Step-by-step explanation:
90 degrees
x + y = 180
x = y
x + x = 180
2x = 180
x = 180/2
x = 90
Both the angle and its supplement = 90
Find the slope of the equation of the line that passes through the given points. through: (0, -3) and (3, 4)
Answer:
7/3
Step-by-step explanation:
The slope is found by dividing the difference in the y-coordinates by the difference in the x-coordinates.
Here, the difference in the y-coordinates is: 4 - (-3) = 4 + 3 = 7. The difference in the x-coordinates is: 3 - 0 = 3.
Dividing 7 by 3, we get a slope of 7/3.
The slope you are looking for is 7.
Write an equation in Slope-Intercept Form for a line that passes through (2, 2) and (0, 3).
What slope would make the lines
parallel?
y = 4x + 2
y = [?]x-4
Enter the number that belongs in the green box.
Answer:
4
Step-by-step explanation:
Parallel lines have the same slope and since the slope of the first line is 4 the second lines slope is also 4
Answer:
Step-by-step explanation:
Slopes of parallel lines are the same.
m = 4
Number 17 thanks pls ASAP ASAP
Answer:
[tex]\frac{1}{2}[/tex] which agrees with answer B
Step-by-step explanation:
First write the equation that represents this type of variation:
[tex]y=\frac{5}{x}[/tex]
then we need to solve for "x" when y = 10 as shown below:
[tex]y=\frac{5}{x} \\10=\frac{5}{x} \\10*x=5\\x=\frac{5}{10} \\x=\frac{1}{2}[/tex]
PLEASE HELP
WHAT ARE THE X AND Y INTERCEPTS!?!?
Answer:
X intercept is the point where the line meets the x. y intercept is the point where the line meets the y. So, the x intercept is -7.5 and the y intercept is 5.5.
Consider the limaçon with equation r = 3 + 4cos(θ). How does the quotient of a and b relate to the existence of an inner loop?
A polar graph that is a limacon has a formula similar to [tex]r=a+bcos\theta[/tex]
Option B is correct.
A limacon has a formula similar to [tex]r=a+bcos\theta[/tex]
Case 1 . If a < b or [tex]\frac{b}{a}>1[/tex]
Then the curve is limacon with an inner loop.
Case 2. If a>b or [tex]\frac{b}{a}<1[/tex]
Then the limacon does not have an inner loop.
Here, given that, [tex]r=3+4cos\theta[/tex]
It is observed that, a < b or [tex]\frac{b}{a}>1[/tex]
Therefore, the curve is limacon with an inner loop.
Hence, option B is correct.
Learn more:
https://brainly.com/question/10381605
Jill purchased 50 game points
at the arcade for $20. If he
wanted to buy 250 game
points, how much would he
spend?
Answer:
He would spend 100 dollars :)!
Hhheleppppp urgggeenttt
Necesito ayuda con una Integral por sustitución trigonométrica, Por favor!
Which expression is equivalent to 6 (a minus 3)?
Answer:
6a-18
Step-by-step explanation:
help help help help help help help help help help
Answer:
27
Step-by-step explanation:
Answer:
the answer its 31
Step-by-step explanation:
and first 21 : (2+1) x 5- 2^2
21 : 3 x 5 -2^2
7x5- 2^2
35-4
31
I hope you understand
What is the solution of the equation?
-39.75 = 7.5m
Answer:
m = -5.3
Step-by-step explanation:
-39.75 = 7.5m
Divide both sides by 7.5 to get m alone
-39.75/7.5 = 7.5m/7.5
-5.3 = m
Hope this helps:)
Answer:
m= -5.3
Step-by-step explanation:
-39.75=7.5m
(divide both sides by 7.5)
-5.3=m
Help someone that knows
A school event requires at least 3 parents for every 10 kids. How many parents for 50 kids?
Answer:
16 i think
Step-by-step explanation:
could i have brain liest tho i 4/5
Answer:
15 parents
Step-by-step explanation:
If there are 3 adults per 10 kids, if there are 50 kids then you would divide 50 by 10 to get five. Then multiply 5 by 3 to get 15.
the product of mais age and 8 is 104
Answer:
mais age is 13
Step-by-step explanation:
104 divided by 8 is 13
HELP ME ASAP ILL GIVE U A BRAINLIEST
Answer:
The pizza was heated 7.50 degrees
The volume of a cylinder is 72 if the radius is 3 what is the height
What additional information could be used to prove that triangle XYZ is congruent to triangle Fe.g. using asa or aas check all that apply
Answer:
its the first one and the last one ..that's the answer.
Draw 3 rows of 10 oranges.
Answer:
what?
how do i draw on here
Step-by-step explanation:
Plz I REALLY NEED HELP
A drawing on a transparency is 6.5 inches wide and 3 inches long. The width of the image of the drawing projected onto a screen is 13 feet. How long, in feet, is the drawing on the screen?
4.) Rotate the image 90 degrees counter-clockwise about the origin. (x,y) -> (-y, x)
Answer:
is rotation of 90° counterclockwise about origin and hence it is the correct answer.Step-by-step explanation:
If f(x) = 2x +4, which of the following is the inverse of f(x)
Step-by-step explanation:
let y = f(x)
y = 2x/7 + 4
2x/7 = y - 4
x/7 = (y - 4)/2
x = 7(y - 4)/2
therefore, f^-1(x) = 7(x - 4)/2 (Option A)
Topic: Functions
If you like to venture further, feel free to check out my insta (learntionary). I'll be constantly posting math tips and notes! Thanks!
I need to know the answers please
how many time does 14 go into 51
because I have to finish if I wont to go watch a new episode of ********
The fifth grade students at East street decided to host a fundraiser for red ribbon week they planed to buy 240 inches of red ribbon how much will it cost them
Answer:
(Amount for 1 inch of red ribbon) x (240) = total cost
Step-by-step explanation:
If the ordered pair (a, b) satisfies the inequality. y > x-4, three of these statements are
true. Which statement is NOT true?
Answer:
If you subtract 4 from a, it will be greater than b
Step-by-step explanation:
Ok, so in order to solve this, it would be easier to substitute (a,b) into the equation:
b>a-4
Ok, next, you just have to work out each individual question to see which one is incorrect. You can immediately know it is the first one, and this is because we know that B is larger in this equation, but the problem says that subtracting 4 from a would be greater than b. This is incorrect, b is greater than 4-a, so the top left would be your answer.
Hope this helps!
What is the most important lab safety rule
Answer:
The most important lab safety rule is to know the location of and how to use safety equipment, such as a fire extinguisher.
Step-by-step explanation:
Hope this helps you
Crown me as brainliest:)
what is 25 divided by 7 expressed as a mixed number
The heights of women in the USA are normally distributed with a mean of 64 inches and a standard deviation of 3 inches.
Required:
a. A random sample of six women is selected. What is the probability that the sample mean is greater than 63 inches?
b. What is the probability that a randomly selected woman is taller than 66 inches?
c. What is the probability that the mean height of a random sample of 100 women is greater than 66 inches?
Answer:
(a) 0.2061
(b) 0.2514
(c) 0
Step-by-step explanation:
Let X denote the heights of women in the USA.
It is provided that X follows a normal distribution with a mean of 64 inches and a standard deviation of 3 inches.
(a)
Compute the probability that the sample mean is greater than 63 inches as follows:
[tex]P(\bar X>63)=P(\frac{\bar X-\mu}{\sigma/\sqrt{n}}>\frac{63-64}{3/\sqrt{6}})\\\\=P(Z>-0.82)\\\\=P(Z<0.82)\\\\=0.20611\\\\\approx 0.2061[/tex]
Thus, the probability that the sample mean is greater than 63 inches is 0.2061.
(b)
Compute the probability that a randomly selected woman is taller than 66 inches as follows:
[tex]P(X>66)=P(\frac{X-\mu}{\sigma}>\frac{66-64}{3})\\\\=P(Z>0.67)\\\\=1-P(Z<0.67)\\\\=1-0.74857\\\\=0.25143\\\\\approx 0.2514[/tex]
Thus, the probability that a randomly selected woman is taller than 66 inches is 0.2514.
(c)
Compute the probability that the mean height of a random sample of 100 women is greater than 66 inches as follows:
[tex]P(\bar X>66)=P(\frac{\bar X-\mu}{\sigma/\sqrt{n}}>\frac{66-64}{3/\sqrt{100}})\\\\=P(Z>6.67)\\\\\ =0[/tex]
Thus, the probability that the mean height of a random sample of 100 women is greater than 66 inches is 0.
