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Nimfa-mama [501]
3 years ago
13

The solution of the system of equations has coordinates where x is between 2 and 3 and y is between 2 and 3. What is a reasonabl

e approximation for each value to the tenths place? x Almost-equals 2.2, y Almost-equals 2.1 x Almost-equals 2.2, y Almost-equals 2.4 x Almost-equals 2.7, y Almost-equals 2.3 x Almost-equals 2.7, y Almost-equals 2.6
Mathematics
1 answer:
g100num [7]3 years ago
8 0

x Almost-equals 2.2, y Almost-equals 2.4

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Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = x4 ln(x) (a) Find the interval on which f is incre
Ainat [17]

Answer: (a) Interval where f is increasing: (0.78,+∞);

Interval where f is decreasing: (0,0.78);

(b) Local minimum: (0.78, - 0.09)

(c) Inflection point: (0.56,-0.06)

Interval concave up: (0.56,+∞)

Interval concave down: (0,0.56)

Step-by-step explanation:

(a) To determine the interval where function f is increasing or decreasing, first derive the function:

f'(x) = \frac{d}{dx}[x^{4}ln(x)]

Using the product rule of derivative, which is: [u(x).v(x)]' = u'(x)v(x) + u(x).v'(x),

you have:

f'(x) = 4x^{3}ln(x) + x_{4}.\frac{1}{x}

f'(x) = 4x^{3}ln(x) + x^{3}

f'(x) = x^{3}[4ln(x) + 1]

Now, find the critical points: f'(x) = 0

x^{3}[4ln(x) + 1] = 0

x^{3} = 0

x = 0

and

4ln(x) + 1 = 0

ln(x) = \frac{-1}{4}

x = e^{\frac{-1}{4} }

x = 0.78

To determine the interval where f(x) is positive (increasing) or negative (decreasing), evaluate the function at each interval:

interval                 x-value                      f'(x)                       result

0<x<0.78                 0.5                 f'(0.5) = -0.22            decreasing

x>0.78                       1                         f'(1) = 1                  increasing

With the table, it can be concluded that in the interval (0,0.78) the function is decreasing while in the interval (0.78, +∞), f is increasing.

Note: As it is a natural logarithm function, there are no negative x-values.

(b) A extremum point (maximum or minimum) is found where f is defined and f' changes signs. In this case:

  • Between 0 and 0.78, the function decreases and at point and it is defined at point 0.78;
  • After 0.78, it increase (has a change of sign) and f is also defined;

Then, x=0.78 is a point of minimum and its y-value is:

f(x) = x^{4}ln(x)

f(0.78) = 0.78^{4}ln(0.78)

f(0.78) = - 0.092

The point of <u>minimum</u> is (0.78, - 0.092)

(c) To determine the inflection point (IP), calculate the second derivative of the function and solve for x:

f"(x) = \frac{d^{2}}{dx^{2}} [x^{3}[4ln(x) + 1]]

f"(x) = 3x^{2}[4ln(x) + 1] + 4x^{2}

f"(x) = x^{2}[12ln(x) + 7]

x^{2}[12ln(x) + 7] = 0

x^{2} = 0\\x = 0

and

12ln(x) + 7 = 0\\ln(x) = \frac{-7}{12} \\x = e^{\frac{-7}{12} }\\x = 0.56

Substituing x in the function:

f(x) = x^{4}ln(x)

f(0.56) = 0.56^{4} ln(0.56)

f(0.56) = - 0.06

The <u>inflection point</u> will be: (0.56, - 0.06)

In a function, the concave is down when f"(x) < 0 and up when f"(x) > 0, adn knowing that the critical points for that derivative are 0 and 0.56:

f"(x) =  x^{2}[12ln(x) + 7]

f"(0.1) = 0.1^{2}[12ln(0.1)+7]

f"(0.1) = - 0.21, i.e. <u>Concave</u> is <u>DOWN.</u>

f"(0.7) = 0.7^{2}[12ln(0.7)+7]

f"(0.7) = + 1.33, i.e. <u>Concave</u> is <u>UP.</u>

4 0
3 years ago
The length of a triangel is three times its with.if the perimiter is at its most 112 centemeters,witch is the greates possible v
iris [78.8K]
I am assuming you do not mean The length of a triangle and you mean The length of a rectangle because the answer you given are the formula for the perimeter of a rectangle. With that said, the answer is D<span>.2w+2*(3w)112

The perimeter of a rectangle can be found by the following equation:

2L + 2W = P
OR 
P = 2L + 2W
The equation </span>P = 2L + 2W is read perimeter = 2 times Length + 2 times width

The Question:
<span>The length of a rectangle is three times its width.if the perimeter is at its most 112 centimeters ,which is the greatest possible value of the the width.
</span>
The length of a rectangle
This means the perimeter 

is
The word is means to use an equal sign

three times its width
This means 3 times w or 3w 
The w = width

perimeter is at its most 112
This means <=
<= means less than or equal to

Ok so lets look at our equation 

2L + 2W = P
2w + 2(3w) <= 112

5 0
3 years ago
In the box there are 20 toys, but 5 of them are broken. Randomly one after another 2 toys are taken out. What are the chances th
vitfil [10]

Answer:

A

Step-by-step explanation:

Refer to the picture above.

Explainations are written in the pic above.

4 0
3 years ago
What is the GCF of 18 and 130?<br> 2,<br> 3,<br> 5,<br> 8,<br> 13,
grandymaker [24]
2 because the others aren’t factors of both
6 0
2 years ago
Read 2 more answers
Which of the following angle pairs best describes angles 6 and 4 in the graphic below?
andriy [413]
Answer: Choice C) Same-side interior angles

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Angle 4 and angle 6 are on the same side, in this case the right hand side of the transversal line (line t). In addition, they are on the interior of the "train tracks" horizontal  lines (line a and line b). Combine this and this is why the two angles are same-side interior angles

Side note: if line a is parallel to line b, then angle 4 and angle 6 add to 180 degrees. At this point, they are considered supplementary.
8 0
3 years ago
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