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Kaylis [27]
3 years ago
9

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

Mathematics
1 answer:
Mariana [72]3 years ago
3 0

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

Check the value of <em>f(x)</em> at these endpoints:

<em>f</em> (0) = 0

<em>f</em> (1) = 0

Take the derivative of <em>f(x)</em> :

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

\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}}

For <em>x</em> ≠ 0, we can eliminate the √<em>x</em> term in the denominator:

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

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

√<em>x</em> (3 - 4<em>x</em>) = 0

√<em>x</em> = 0   <u>or</u>   3 - 4<em>x</em> = 0

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

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he amount of time that a customer spends waiting at an airport check-in counter is a random variable with mean 8.3 minutes and s
sp2606 [1]

Complete question:

He amount of time that a customer spends waiting at an airport check-in counter is a random variable with mean 8.3 minutes and standard deviation 1.4 minutes. Suppose that a random sample of n equals 47 customers is observed. Find the probability that the average time waiting in line for these customers is

a) less than 8 minutes

b) between 8 and 9 minutes

c) less than 7.5 minutes

Answer:

a) 0.0708

b) 0.9291

c) 0.0000

Step-by-step explanation:

Given:

n = 47

u = 8.3 mins

s.d = 1.4 mins

a) Less than 8 minutes:

P(X

P(X' < 8) = P(Z< - 1.47)

Using the normal distribution table:

NORMSDIST(-1.47)

= 0.0708

b) between 8 and 9 minutes:

P(8< X' <9) =[\frac{8-8.3}{1.4/ \sqrt{47}}< \frac{X'-u}{s.d/ \sqrt{n}} < \frac{9-8.3}{1.4/ \sqrt{47}}]

= P(-1.47 <Z< 6.366)

= P( Z< 6.366) - P(Z< -1.47)

Using normal distribution table,

NORMSDIST(6.366)-NORMSDIST(-1.47)

0.9999 - 0.0708

= 0.9291

c) Less than 7.5 minutes:

P(X'<7.5) = P [Z< \frac{7.5-8.3}{1.4/ \sqrt{47}}]

P(X' < 7.5) = P(Z< -3.92)

NORMSDIST (-3.92)

= 0.0000

3 0
3 years ago
the functuon y=30+5x represents the cost y (in dollars) of having your dog groomed and buying x extra services. does this situat
Sveta_85 [38]

We want to see if the given function represents a linear function.

We will see that yes, it does.

So we have the function:

y = f(x) = 30 + 5*x

And a general linear function is written as:

y = a*x + b

where a is the slope and b is the y-intercept.

Comparing these two, we can see that both have the same general shape, thus, our function is also a linear function.

A graph of our function also can be seen below, where it is evident that it is a line.

If you want to learn more, you can read:

brainly.com/question/20286983

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Simplify the expression. Explain each step 2+(a+8)
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Answer:

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Step-by-step explanation:

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n = a number

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