F(x)=8x sqrt(x-x^2) Find the exact maximum

First take note of the domain of f(x) ; the square root term is defined as long as x - x ² ≥ 0, or 0 ≤ x ≤ 1.

Check the value of f(x) at these endpoints:

f (0) = 0

f (1) = 0

Take the derivative of f(x) :

[tex]f(x)=8x\sqrt{x-x^2}=8x\left(x-x^2\right)^{\frac12}[/tex]

[tex]\implies f'(x)=8\left(x-x^2\right)^{\frac12}+4x\left(x-x^2\right)^{-\frac12}(1-2x)=4\left(x-x^2\right)^{-\frac12}\left(2\left(x-x^2)\right)+x(1-2x)\right)=\dfrac{4(3x-4x^2)}{\sqrt{x-x^2}}[/tex]

For x ≠ 0, we can eliminate the √x term in the denominator:

[tex]x\neq0\implies f'(x)=\dfrac{4\sqrt x (3-4x)}{\sqrt{1-x}}[/tex]

f(x) has critical points where f '(x) is zero or undefined. We know about the undefined case, which occurs at the boundary of the domain of f(x). Check where f '(x) = 0 :

√x (3 - 4x) = 0

√x = 0   or   3 - 4x = 0

The first case gives x = 0, which we ignore. The second leaves us with x = 3/4, at which point we get a maximum of max{f(x) } = 3√3 / 2.

Three percent of the students at Midwood High School are left-handed. There are 700 students at Midwood High School. How many students at Midwood High School are left-handed?
A.21 students
B.210 students
C.140 students
D.12 students

Find the value of x

Nicole received the following scores on her last 10 math assignments.

А Campus athletics is interested in the proportion of registered students living on campus who play basketball recreationally during the semester. They take a random sample of 198 registered students listed as living in campus housing and ask each whether or not they do so. A 95% confidence interval is correctly calculated from this data as (0.12, 0.18). Which of the following is the correct interpretation of this confidence interval ?
a. None of these are correct.
b. We are 95% confident that the proportion of the 198 sampled students that actually play basketball is between 0.12 and 0.18
c. We are 95% confident that, among the population of students that play basketball recreationally, the proportion who live on campus is between 0.12 and 0.18.
d. The sample proportion of on-campus students that play basketball will be between 0.12 and 0.18 in 95% of samples of size 198

WHAT IS 3/20 IN DECIMAL AND PERCENTAGE

The product of 2 and 1 and 1/2 is ______ 2.


Greater than
Equal to
Less than
No answer text provided.

no link plz

Me.Bernstein will cut 8 pies into pieces that are each 1/6 of the whole After he cuts the 8 pies. how many pieces will Mr.Bernstein have

Amy needs to estimate how many students at her school like online testing. She needs to create a random sample of students. How should Amy collect her sample? Select two that apply.
O Amy should ask every 10th person leaving the school at the end of the day.
O Amy should ask 3 people at every table in the cafeteria
O Amy should ask her 10 closest friends.
O Amy should ask the 35 students in her honors math class
O Amy should ask the 50 students in the computer club.​

) Cars are parked in line as they come off the assembly lines. There are three models: red cars which take up 2 spaces, blue cars which also take up 2 spaces, and green cars which take up only 1 space. Let an be the number of ways of filling the first n parking spaces with red, blue and green cars.

I will give brainlyist and extra points for correct answer ( no links )

Paisley is going to invest $28,000 and leave it in an account for 14 years. Assuming the interest is compounded monthly, what interest rate, to the nearest tenth of a percent, would be required in order for Paisley to end up with $74,000?

What is the unit rate for the following rate: 5 pounds of potatoes costs $10

log 3x^2y fully expanded

A pride of lions is growing at a rate of 0.5% per year, compounded continuously. If the growth rate continues, how large will
the pride be after 100 years if it starts with 50 lions. Round your answer to the nearest lion.
Provide your answer below:

Name: LESTER GREEN Class: NIGHT SCHOOL Complete with the missing letters and then colour. ED BAK _LU GRE_ W_ _TE PN B_O_N GR. N. a ELL__ RE_ OR N E PURP__ i havourite Colours​

Evaluate the expression for x = 2 and y = 4 . (x ^ 5 * y ^ 7) / (x ^ 3 * y ^ 5) numbers only

15 is 9 more than j.
As an equation.

Simplify: (3^5)^2 / 3^-2


A) 3^10

B) 3^9

C) 3^12

D) 3^8

Please help will give brainliest

Solve the following equation: *

A car journey is in two stages Stage1 the car travels 110 miles in 2 hours Stage 2 the car travels 44 miles at the same average speed as stage 1 Work out the time for stage 2 Give you answer in minutes.

In which step did the student make the first error?
step 1
step 2
step 3
step 4

How to figure out square feet of counter tops

What is the area of the school crossing sign represented below?
15 inches 15 inches
15 inches
15 inches
help me plsss

Who ever answers it gets Brainly

Which of the following equation represents a linear function

X^3 - 5x^2 + 2x + 8 = 0; x - 2 factor and find all roots

Need the answer for x and y

1. Which of the following is not a true statement?

Group of answer choices

Angle 1 and Angle 3 are supplementary angles.

Angle 2 and Angle 7 are congruent

Angle 5 and Angle 8 are congruent.​

Angle 3 and Angle 7 are supplementary angles.

I need some help!! I wil try to give brainliest!!!

Answer choices- 4, 64, 240, 116, 68, 105, 480, 400 (20 points)​