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salantis [7]
3 years ago
11

Writey=1/8 x+7 in standard form using integers.

Mathematics
1 answer:
sineoko [7]3 years ago
5 0

Answer:

x - 8y = - 56

Step-by-step explanation:

The equation of a line in standard form is

Ax + By = C ( A is a positive integer and B, C are integers )

Given

y = \frac{1}{8} x + 7

Multiply through by 8 to clear the fraction

8y = x + 56 ( subtract 8y from both sides )

0 = x - 8y + 56 ( subtract 56 from both sides )

- 56 = x - 8y, that is

x - 8y = - 56 ← in standard form

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Consider the equation below. (If you need to use -[infinity] or [infinity], enter -INFINITY or INFINITY.)f(x) = 2x3 + 3x2 − 180x
soldier1979 [14.2K]

Answer:

(a) The function is increasing \left(-\infty, -6\right) \cup \left(5, \infty\right) and decreasing \left(-6, 5\right)

(b) The local minimum is x = 5 and the maximum is x = -6

(c) The inflection point is x = -\frac{1}{2}

(d) The function is concave upward on \left(- \frac{1}{2}, \infty\right) and concave downward on \left(-\infty, - \frac{1}{2}\right)

Step-by-step explanation:

(a) To find the intervals where f(x) = 2x^3 + 3x^2 -180x is increasing or decreasing you must:

1. Differentiate the function

\frac{d}{dx}f(x) =\frac{d}{dx}(2x^3 + 3x^2 -180x) \\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\f'(x)=\frac{d}{dx}\left(2x^3\right)+\frac{d}{dx}\left(3x^2\right)-\frac{d}{dx}\left(180x\right)\\\\f'(x) =6x^2+6x-180

2. Now we want to find the intervals where f'(x) is positive or negative. This is done using critical points, which are the points where f'(x) is either 0 or undefined.

f'(x) =6x^2+6x-180 =0\\\\6x^2+6x-180 = 6\left(x-5\right)\left(x+6\right)=0\\\\x=5,\:x=-6

These points divide the number line into three intervals:

(-\infty,-6), (-6,5), and (5, \infty)

Evaluate f'(x) at each interval to see if it's positive or negative on that interval.

\left\begin{array}{cccc}Interval&x-value&f'(x)&Verdict\\(-\infty,-6)&-7&72&Increasing\\(-6,5)&0&-180&Decreasing\\(5, \infty)&6&72&Increasing\end{array}\right

Therefore f(x) is increasing \left(-\infty, -6\right) \cup \left(5, \infty\right) and decreasing \left(-6, 5\right)

(b) Now that we know the intervals where f(x) increases or decreases, we can find its extremum points. An extremum point would be a point where f(x) is defined and f'(x) changes signs.

We know that:

  • f(x) increases before x = -6, decreases after it, and is defined at x = -6. So f(x) has a relative maximum point at x = -6.
  • f(x) decreases before x = 5, increases after it, and is defined at x = 5. So f(x) has a relative minimum point at x = 5.

(c)-(d) An Inflection Point is where a curve changes from Concave upward to Concave downward (or vice versa).

Concave upward is when the slope increases and concave downward is when the slope decreases.

To find the inflection points of f(x), we need to use the f''(x)

f''(x)=\frac{d}{dx}\left(6x^2+6x-180\right)\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\f''(x)=\frac{d}{dx}\left(6x^2\right)+\frac{d}{dx}\left(6x\right)-\frac{d}{dx}\left(180\right)\\\\f''(x) =12x+6

We set f''(x) = 0

f''(x) =12x+6 =0\\\\x=-\frac{1}{2}

Analyzing concavity, we get

\left\begin{array}{cccc}Interval&x-value&f''(x)\\(-\infty,-1/2)&-2&-18\\(-1/2,\infty)&0&6\\\end{array}\right

The function is concave upward on (-1/2,\infty) because the f''(x) > 0 and concave downward on (-\infty,-1/2) because the f''(x) < 0.

f(x) is concave down before x = -\frac{1}{2}, concave up after it. So f(x) has an inflection point at x = -\frac{1}{2}.

7 0
3 years ago
Someone please help!!! Due tonight, I’ll give brainliest!!!
Ad libitum [116K]

Answer:

2 * 0.75 + 0.50 + 4 * 1.25

$7.00

Step-by-step explanation:

Expression

2 * 0.75 + 0.50 + 4 * 1.25

Evaluate

1.50 + 0.50 + 5.00

$7.00

6 0
3 years ago
A high-speed computer printer prints a page in 1/6 second. Using this printer, how long would it take to print 30 pages?
german
30*\frac{1}{6}=\frac{30}{6}=5
6 0
3 years ago
Read 2 more answers
Laura is planning a party for her son. She has $50 dollars remaining in her budget and wants to provide one party favor per pers
enyata [817]

Answer:

The system of inequalities that represents this situation is:

x+y≥10

6x+4y≤50

Step-by-step explanation:

As the statement says that Laura wants to provide one party favor per person to at least 10 guests, the first inequality would indicate that the number of stuffed animals plus the number of toy trucks should be equal or greater than 10:

x+y≥10

Also, the statement indicates that miniature stuffed animals cost $6.00 each and the toy trucks cost $4.00 each and that Laura has $50. From this, you would have an inequality that indicates that 6 for the number of miniature stuffed animals and 4 for the number of toy trucks would be equal or less than 50:

6x+4y≤50

The answer is that the system of inequalities that represents this situation is:

x+y≥10

6x+4y≤50

4 0
3 years ago
These are questions 2-5 out of 20. are these correct!?
dusya [7]
I believe so, but It is kind of blurry for some reason on my screen but from what I can make out yes.
3 0
3 years ago
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