Let f(x)=4−frac{2}{x}+frac{6}{x^{2}}. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima). f is increasing on the intervals f is decreasing on the intervals The relative maxima of f occur at x = The relative minima of f occur at x = Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word "none". In the last two, your answer should be a comma separated list of x values or the word "none".

Let f(x)=4−frac{2}{x}+frac{6}{x^{2}}. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima). f is increasing on the intervals f is decreasing on the intervals The relative maxima of f occur at x = The relative minima of f occur at x = Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word "none". In the last two, your answer should be a comma separated list of x values or the word "none".

Question
Confidence intervals
asked 2021-03-11
Let \(f(x)=4−\frac{2}{x}+\frac{6}{x^{2}}\).
Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima).
f is increasing on the intervals
f is decreasing on the intervals
The relative maxima of f occur at x =
The relative minima of f occur at x =
Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word "none".
In the last two, your answer should be a comma separated list of x values or the word "none".

Answers (1)

2021-03-12
\(f(x) = 4 - \frac{2}{x}+\frac{6}{x^{2}}\)
Take derivative and find out critical points to get maxima and minima
Apply power rule
\(f(x) = 4 - \frac{2}{x}+\frac{6}{x^{2}}\)
\(f`(x) = 0 + \frac{2}{x^{2}}+\frac{12}{x^{3}}\)
\(f`(x) = \frac{2}{x^{2}}+\frac{12}{x^{3}}\)
Set the derivative \(=0\) and solve for x
LCD is \(x^{3}\),
multiply each term by \(x^{3}\)
\(\frac{2}{x^{2}}-\frac{12}{x^{3}}=0\)
\(\frac{2}{x^{2}}(x^{3})-\frac{12}{x^{3}}(x^{3})=0(x^{3})\)
\(2x - 12 = 0\)
\(2x = 12\)
\(x = 6\)
\(x=0\) makes denominator 0 and it is undefined
So \(x=0, x=6\)
Break the numbers line into three intervals using 0 and 6
Check each interval using derivative Let \(x=-1 , x=1, x=7\)
\(f'(x)=\frac{2}{x^{2}}-\frac{12}{x^{3}}\)
\(f'(-1)=\frac{2}{(1)^{2}}-\frac{12}{(1)^{3}}=-10\)
\(f'(1)=\frac{2}{(1)^{2}}-\frac{12}{(1)^{3}}=-10\)
\(f'(7)=\frac{2}{(7)^{2}}-\frac{12}{(7)^{3}}=\frac{2}{343}\)
Derivative is from positive in two intervals
So increasing intervals are\((-\infty,0) U(6, \infty)\)
Decreasing interval \((0, 6)\)
There is a break in graph of \(f(x) at x = 0\)
The derivative goes from negative to positive at \(x = 6\)
So relative minima at \(x = 6\)
There is no relative maxima
0

Relevant Questions

asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
asked 2021-05-05

A random sample of \( n_1 = 14 \) winter days in Denver gave a sample mean pollution index \( x_1 = 43 \).
Previous studies show that \( \sigma_1 = 19 \).
For Englewood (a suburb of Denver), a random sample of \( n_2 = 12 \) winter days gave a sample mean pollution index of \( x_2 = 37 \).
Previous studies show that \( \sigma_2 = 13 \).
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
\( H_0:\mu_1=\mu_2.\mu_1>\mu_2 \)
\( H_0:\mu_1<\mu_2.\mu_1=\mu_2 \)
\( H_0:\mu_1=\mu_2.\mu_1<\mu_2 \)
\( H_0:\mu_1=\mu_2.\mu_1\neq\mu_2 \)
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference \( \mu_1 - \mu_2 \). Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the \( \alpha = 0.01 \) level, we reject the null hypothesis and conclude the data are statistically significant.
At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the \( \alpha = 0.01 \) level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
\( \mu_1 - \mu_2 \).
(Round your answers to two decimal places.)
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
asked 2020-10-18
Consider the following function. \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}^{{{2}}}}}{{{x}^{{{2}}}-{81}}}}\) a) To find the critucal numbers of f. b) To find the open interval on which function is increasing or decreasing. c) To identify the relative extremum.
asked 2021-04-11
The equation F=−vex(dm/dt) for the thrust on a rocket, can also be applied to an airplane propeller. In fact, there are two contributions to the thrust: one positive and one negative. The positive contribution comes from air pushed backward, away from the propeller (so dm/dt<0), at a speed vex relative to the propeller. The negative contribution comes from this same quantity of air flowing into the front of the propeller (so dm/dt>0) at speed v, equal to the speed of the airplane through the air.
For a Cessna 182 (a single-engine airplane) flying at 130 km/h, 150 kg of air flows through the propeller each second and the propeller develops a net thrust of 1300 N. Determine the speed increase (in km/h) that the propeller imparts to the air.
asked 2021-04-21
Car 1 has a mass of m1 = 65 ❝ 103 kg and moves at a velocity of v01 = +0.81 m/s. Car 2, with a mass of m2 = 92 ❝ 103 kg and a velocity of v02 = +1.2 m/s, overtakes car 1 and couples to it. Neglect the effects of friction in your answer.
(a) Determine the velocity of their center of mass before the collision m/s
(b) Determine the velocity of their center of mass after the collision m/s
(c) Should your answer in part (b) be less than, greater than, or equal to the common velocity vf of the two coupled cars after the collision? less than greater than equal to
asked 2021-05-18
The student engineer of a campus radio station wishes to verify the effectivencess of the lightning rod on the antenna mast. The unknown resistance \(\displaystyle{R}_{{x}}\) is between points C and E. Point E is a "true ground", but is inaccessible for direct measurement because the stratum in which it is located is several meters below Earth's surface. Two identical rods are driven into the ground at A and B, introducing an unknown resistance \(\displaystyle{R}_{{y}}\). The procedure for finding the unknown resistance \(\displaystyle{R}_{{x}}\) is as follows. Measure resistance \(\displaystyle{R}_{{1}}\) between points A and B. Then connect A and B with a heavy conducting wire and measure resistance \(\displaystyle{R}_{{2}}\) between points A and C.Derive a formula for \(\displaystyle{R}_{{x}}\) in terms of the observable resistances \(\displaystyle{R}_{{1}}\) and \(\displaystyle{R}_{{2}}\). A satisfactory ground resistance would be \(\displaystyle{R}_{{x}}{<}{2.0}\) Ohms. Is the grounding of the station adequate if measurments give \(\displaystyle{R}_{{1}}={13}{O}{h}{m}{s}\) and R_2=6.0 Ohms?
asked 2021-05-07
Let f be a periodic function with period 2 which is given on the interval (-1,1) by f(x)=x. Find all distributional derivatives \(\displaystyle{{D}_{{{x}}}^{{{m}}}}{f}\).
asked 2021-04-15
A car initially traveling eastward turns north by traveling in a circular path at uniform speed as in the figure below. The length of the arc ABC is 235 m, and the car completes the turn in 33.0 s. (Enter only the answers in the input boxes separately given.)
(a) What is the acceleration when the car is at B located at an angle of 35.0°? Express your answer in terms of the unit vectors \(\displaystyle\hat{{{i}}}\) and \(\displaystyle\hat{{{j}}}\).
1. (Enter in box 1) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+{\left({E}{n}{t}{e}{r}\in{b}\otimes{2}\right)}{P}{S}{K}\frac{{m}}{{s}^{{2}}}\hat{{{j}}}\)
(b) Determine the car's average speed.
3. ( Enter in box 3) m/s
(c) Determine its average acceleration during the 33.0-s interval.
4. ( Enter in box 4) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+\)
5. ( Enter in box 5) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{j}}}\)
asked 2021-04-25
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius \(\displaystyle{R}={7.4}\times{10}^{{-{15}}}\) m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit \(\displaystyle={1.66}\times{10}^{{-{27}}}\) kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
asked 2021-05-03
A charge of \(\displaystyle{6.00}\times{10}^{{-{9}}}\) C and a charge of \(\displaystyle-{3.00}\times{10}^{{-{9}}}\) C are separated by a distance of 60.0 cm. Find the position at which a third charge, of \(\displaystyle{12.0}\times{10}^{{-{9}}}\) C, can be placed so that the net electrostatic force on it is zero.
...