Use the First Derivative Test to determine whether the critical point is a local min or max (or neither). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x)=(ln(2x)/8x)-7 (x>0)
Also determine the intervals on which the function is increasing or decreasing.
Answers
Answer:
Step-by-step explanation:
Let be [tex]f(x) = \frac{1}{8\cdot x}\cdot \ln(2\cdot x) -7[/tex], the first and second derivatives of the function are, respectively:
[tex]f'(x) = \frac{\left(\frac{2}{2\cdot x}\right)\cdot 8\cdot x-8\cdot \ln (2\cdot x) }{64\cdot x^{2}}[/tex]
[tex]f'(x) = \frac{8-8\cdot \ln (2\cdot x)}{64\cdot x^{2}}[/tex]
[tex]f'(x) = \frac{1-\ln(2\cdot x)}{8\cdot x^{2}}[/tex]
[tex]f''(x) = \frac{\left(-\frac{2}{2\cdot x} \right)\cdot (8\cdot x^{2})-[1-\ln(2\cdot x)]\cdot (16\cdot x)}{64\cdot x^{4}}[/tex]
[tex]f''(x) = \frac{-8\cdot x-16\cdot x+16\cdot x\cdot \ln (2\cdot x)}{64\cdot x^{4}}[/tex]
[tex]f''(x) = \frac{-24\cdot x+16\cdot x \cdot \ln (2\cdot x)}{64\cdot x^{4}}[/tex]
[tex]f''(x) = -\frac{3}{8\cdot x^{3}}+\frac{\ln (2\cdot x)}{4\cdot x^{3}}[/tex]
Now, let equalise the first derivative to zero and solve the resulting expression:
[tex]\frac{1-\ln(2\cdot x)}{8\cdot x^{2}} = 0[/tex]
[tex]1-\ln(2\cdot x) = 0[/tex]
[tex]\ln(2\cdot x) = 1[/tex]
[tex]\ln 2 +\ln x = 1[/tex]
[tex]\ln x = 1-\ln 2[/tex]
[tex]x = e^{1-\ln 2}[/tex]
[tex]x = \frac{e}{e^{\ln 2}}[/tex]
[tex]x = \frac{e}{2}[/tex]
[tex]x \approx 1.359[/tex]
This result is evaluated at the second derivative expression:
[tex]f''(1.359) =-\frac{3}{8\cdot (1.359)^{3}}+\frac{\ln [2\cdot (1.359)]}{4\cdot (1.359)^{3}}[/tex]
[tex]f''(1.359)\approx -0.050[/tex]
The critical value leads to a critical maximum and there are two intervals:
[tex](0, 1.359)[/tex] - Increasing
[tex](1.359,+\infty )[/tex] - Decreasing
The graphic of the function is presented below as attachment.
Related Questions
Fill in the blanks to simplify the expression. 3{4 × [3+ (1+2 × 5)]} = 3{4× [3+ □] } =3{□} = □.
Answers
Answer: 11, 56, 168
Step-by-step explanation: order of operations:
1+10=11.
3+11=14. 14x4=56.
3x56=168
Round 7/12 to the nearest hundredth
Answers
Answer:
7.00
Step-by-step explanation:
so you have to round to the narest 0.01 the easy way to do it is round to the hundreth place. And you will get your answer which is 7.00
Ben can type 25 words in half a minute at the rate how many words can he type in 5 minutes
Answers
Answer: 250 words in 5 minutes
Step-by-step explanation:
Find the value of x in the equation below. 2x+8/6=1/3(x+4) A. x = 0 B. no real solutions C. infinitely many solutions D. x = 1
Answers
Answer:
x = 0
Step-by-step explanation:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
2*x+8/6-(1/3*(x+4))=0
Simplify 1/3
Equation at the end of step 1 :
(2x + 8/6) - (1/3 * (x + 4)) = 0
STEP 2 :
Equation at the end of step 2
(2x + 8/6) - (x+4)/3 = 0
STEP 3 :
Simplify 4/3
Equation at the end of step 3 :
(2x + 4/3) - (x+4)/3 = 0
STEP 4 :
Rewriting the whole as an Equivalent Fraction
Adding a fraction to a whole
Rewrite the whole as a fraction using 3 as the denominator :
2x = 2x/1 = 2x*3/3
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
2x * 3 + 4/3 = 6x+4/3
Equation at the end of step 4 :
(6x+4)/3 - (x+4)/3 = 0
STEP 5 : Pulling out like terms
Pull out like factors :
6x + 4 = 2*(3x + 2)
Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
2*(3x+2)-((x+4))/3 = 5x/3
Equation at the end of step 5 :
5x/3 = 0
STEP 6: When a fraction equals zero
Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.
Now, to get rid of the denominator, multiply both sides of the equation by the denominator.
Here's how:
5x/3*3 = 0*3
Now, on the left hand side, the 3 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
5x = 0
Solving a Single Variable Equation:
Solve : 5x = 0
Divide both sides of the equation by 5:
x = 0
for what values of x will 4^x be greater than 1 ?
Answers
Answer:
x>0
Step-by-step explanation:
We want to solve for x in the inequality:
4^x > 1
ln(4^x)=xln(4) > ln(1)=0
x > 0/ln(4) =0
x>0
So for any value of x greater than 0, you will have 4^x be greater than 1.
There are 180 girls in a mixed school. if the ratio of girls to Boyd is 4:3,find the total number of students in the school.
Answers
Step 1) Set up a proportion
180 girls / x boys = 4 girls / 3 boys
540 = 4x
x = 135 boys
Step 2) Add the number of boys and girls together
180 + 135 = 315
Answer: 315 total students
No sé cómo hacerlo. Y me gustaría que me ayudaran con él proceso?
Answers
Answer:
BC=11
Step-by-step explanation:
we need to find BC
and we know that
AB= x+2
AC=13
BC=2x+11
A, B and C are collinear
that means that
AB+BC=AC
x+2+2x+11=13
3x+13=13
3x=0
x=0
so BC=2(0)+11
BC=11
in abc the measure of side c is 3.9cm if def has a dilation of abc with a scale factor of 2.5
Answers
Complete Question:
In ABC, the measure of sides c is 3.9 cm If DEF is a dilation ABC with a scale factor of 2.5 what is the measure of side f
Answer:
f = 9.75
Step-by-step Explanation:
From the information given, a sketch of ∆ABC and ∆DEF with their corresponding sides and angles have been attached below.
∆DEF is a dilation of ∆ABC, which is said to be on a scale factor of 2.5.
The scale factor is a whole number, this implies that ∆DEF is an enlargement of ∆ABC.
Since side c = 3.9 cm, in ∆ABC corresponds to side f, in ∆DEF, therefore, the measure of f would be:
f = measure of c × scale factor
f = 3.9 cm × 2.5
f = 9.75
Let
f(x) =
7x − 5, x ≤ −5
−x2, x > −5
.
Find
f(−1), f(−5), and f(−6)
Answers
Answer:
EZ
Step-by-step explanation:
EZ
a(x+b)=g+3f
solve for x
please help<333
Answers
Answer a=3f+g/b+x
Step-by-step explanation:
Let's solve for a.
a(x+b)=g+3f
Step 1: Factor out variable a.
a(b+x)=3f+g
Step 2: Divide both sides by b+x.
a(b+x)/b+x=3f+g/b+x
a=3f+g/b+x
The park wanted to put up a new 500 foot fence around the rectangular playground. The length of the fence is going to be 100 foot. what will be the with of the fence?
Answers
Answer:
150 feet
Step-by-step explanation:
A rectangle has 2 pairs of equal sides.
One of the sides is 100 feet, meaning another side is 100 feet.
If we subtract 200 feet we are left with 300, which are the last two sides.
Divide 300 by 2 to get each side length.
150 feet
Find an equation of the line passing through the points (−8,2) with the slope m= 4/3
Answers
Answer:
[tex]y=\frac{4}{3}x+\frac{38}{3}[/tex]
Step-by-step explanation:
So we want to find an equation of a line passing through (-8,2) with a slope of 4/3.
To do so, we can use the point-slope form. This is:
[tex]y-y_1=m(x-x_1)[/tex]
So, substitute (-8,2) for x₁ and y₁, and let m be 4/3:
[tex]y-2=\frac{4}{3}(x+8)[/tex]
Distribute the right:
[tex]y-2=\frac{4}{3}x+\frac{32}{3}[/tex]
Add 2 to both sides. Note that 2 can be written as 6/3. Thus:
[tex]y=\frac{4}{3}x+\frac{32}{3}+\frac{6}{3}[/tex]
Simplify:
[tex]y=\frac{4}{3}x+\frac{38}{3}[/tex]
And this is our equation :)
What is the answer rounded to the nearest whole number
Answers
What’s 2 1/6 - 1 5/6?
Answers
Answer:
2/6 or 1/3
Step-by-step explanation:
please assist me with this problem: part 10
Answers
Answer:
c
Step-by-step explanation:
Given
sinN = [tex]\frac{a}{b}[/tex] ( multiply both sides by b ) , then
a = bsinN → (a)
And
sinM = [tex]\frac{a}{y}[/tex] ( multiply both sides by y ), then
a = ysinM → (b)
Thus a is both (a) and (b)
Two pieces of fabric
are 143/4 inches and
207/8 inches long. To
form a single piece in
length, 3/8 inch of
each masonry piece
was sewn together to
create a seam. What
is the final length of
the fabric after the
two pieces are sewn?
Answers
Answer:
487/8 inchesStep-by-step explanation:
[tex]\left(\dfrac{143}{4}-\dfrac38\right)+\left(\dfrac{207}{8}-\dfrac38\right)=\dfrac{286}{8}-\dfrac38+\dfrac{204}{8}=\dfrac{283}{8}+\dfrac{204}{8}=\dfrac{487}{8}[/tex]
PLEASE HELP !
Expand the following numbers ;
1.) 1.23 x 10^0
2.) 1.54 x 10^4
3.) 2.5 x 10^-3
4.) 5.67 x 10^-1
5.) 1.00 x 10^8
6.) 1.00 x 10^-8
Answers
Answer:
1. 1.23
2. 15400
3. 0.0025
4. 0.567
5. 100000000
6. 0.00000001
Step-by-step explanation:
you just move the decimal point left for negative and right for positive
(Anything to the power of 0 = 1)
2) 1.54 x 10000 = 15400
3) 2.5 x 0.001 = 0.0025
4) 5.67 x 0.1 = 0.567
5) 1 x 100000000 = 100000000
6) 1 x 0.00000001 = 0.00000001
Refer to the sample data for pre-employment drug screening shown below. If one of the subjects is randomly selected, what is the probability that the test result is a false positive? Who would suffer from a false positive result? Why?
Pre-Employment Drug Screening Results
Positive test result Negative test result
Drug Use Is Indicated Drug Use Is Not Indicated
Subject Uses Drugs 38 12
Subject Is Not a drug user 19 29
Answers
Answer:
0.193877
Step-by-step explanation:
The data given to us is
Pre-Employment Drug Screening Results
Positive test result Negative test result
Drug Use Is Indicated Drug Use Is Not Indicated
Subject Uses Drugs: 38 12
Subject Is Not a drug user: 19 29
Now the total of this is = 38+19+12+29= 98
Now the probability of false positive is = 19/98= 0.193877
The Subject Is Not a drug user would suffer from a false positive. He is not a user and has a positive result.
7) A jet travels 440 miles in 2 hours. At this rate, how far could the jet Hy
12 hours? What is the rate of speed of the jet?
Answers
Answer:
2,640 miles in 12 hours.
Step-by-step explanation:
440/2=220mph
220*12 hours=2,640
rate of speed=220pmh
So 220 mph
simplify the algebraic expression 3xy+5x+2+3y+x+4
Answers
Answer:
3xy+6x+3y+6
Step-by-step explanation:
You just need to combine like terms.
5x and x are like terms so you can add them to 6x.
2 and 4 are like terms so you can add them to 6
Find the area to the nearest square foot of the shaded region below, consisting of a square with a circle cut out of it. Use 3.14 as an approximation for π. Need this ASAP
Answers
Answer:
22 square feet
Step-by-step explanation:
Area of shaded region = Area of square - Area of circle. The diameter of the circle is also the length of the side of the square as the circumference of the circle is on the circumference of the square.
Area of circle = 3.14 × r × r
r = 10 ÷ 2
= 5
Area of circle = 3.14 × 5 × 5
= 78.5
Area of square = length × length
Area of square = 10 × 10
= 100
Area of shaded region = Square - Circle
= 100 - 78.5
= 21.5
= 22 (to the nearest square foot)
I hope this helps :)
Help it’s not b what is the answer plz help
Answers
Answer:
c
Step-by-step explanation:
Please see attached picture for full solution.
At a sporting goods store, Franklin tennis racquets normally sell for $110, but this week
they're selling for less than their normal price, How much has the price on each
d'acquet been lowered?
Answers
20 POINTSS!! Choose the fraction(s) equivalent to the given fraction. -1/7 Select all that apply A. -1/7 B. 1/-7 C. 1/7 D. -1/-7
Answers
Answer:
A and B.
Step-by-step explanation:
So we want to select the fractions that equal -1/7.
Let's go through each of the answer choices.
A)
A is -1/7, the exact same.
A is correct.
B)
B is 1/-7.
We can move the negative to the top.
So, 1/-7 is equivalent to -1/7.
B is also correct.
C)
C is 1/7
There are no negatives whatsoever.
C is not correct.
D)
We have -1/-7.
Again, move the negative to the top. We will have:
-(-1)/7
The two negatives cancel:
=1/7.
So, D is not correct.
Our answers are A and B.
English
A section of wall is being framed. A model of the framing
work is shown below.
Which best describes the relationship between the
125° angle and angle A?
They are same side interior angles. Angle A
measures 55°
They are alternate interior angles. Angle A
measures 125
They are vertical angles. Angle A measures 125°.
They are corresponding angles. Angle A measures
55°
1250
b
e
Intro
Done
Answers
Answer:
They are some side interior angles. Angle A measures 55 degrees.
Step-by-step explanation:
Angle A is accuse so it can’t be 90 or more, so that narrows you’re options down by two. From there it’s fairly easy to distinguish.
The property tax on a house with an assessed value of $480,000 is $5760. Determine the property tax on a house with an assessed value of $600,000, assuming the same tax rate.
Answers
Answer:
Hey there!
5760/480000=x/600000
Solving for x, we get the answer is $7200.
Let me know if this helps :)
Answer:
$7200
Step-by-step explanation:
We can start by finding out the tax rate by dividing the property tax by assessed value.
$5760/$480,000 = 0.012
From this, we learn the tax rate is 1.2%
Now, we can multiply the assessed value of $600,000 by the tax rate.
$600,000*0.012 = $7200
The property tax on a house with the assessed value of $600,000 is $7200.
Pleaseeeeeeee helpppp
Answers
Answer:
point G
Step-by-step explanation:
Find the referenced points on the diagram and notice the lines between them. The point at the intersection of those lines is what is wanted.
It is point G.
The sum of the sequence of 685+678+671+664+...+6
Answers
Answer:
Sum of the sequence (Sn) = 33,859
Step-by-step explanation:
Given:
Sequence = 685+678+671+664+...+6
Find:
Sum of the sequence (Sn)
Computation:
a = 685
d = 678 - 985 = -7
an = 6
an = a+(n-1)d
6 = 685+(n-1)(-7)
-679 = (n-1)(-7)
97 = n-1
n = 98
So,
Sum of the sequence (Sn) = (n/2)[a+an]
Sum of the sequence (Sn) = (98/2)[685+6]
Sum of the sequence (Sn) = (49)(691)
Sum of the sequence (Sn) = 33,859
6x- 10=-8+3x
Solve the linear equation.
Answers
Answer:
x =2/3Step-by-step explanation:
[tex]6x-10=-8+3x\\\\\mathrm{Add\:}10\mathrm{\:to\:both\:sides}\\6x-10+10=-8+3x+10\\\\\mathrm{Simplify}\\6x=3x+2\\\\\mathrm{Subtract\:}3x\mathrm{\:from\:both\:sides}\\6x-3x=3x+2-3x\\\\Simplify\\3x=2\\\\\mathrm{Divide\:both\:sides\:by\:}3\\\frac{3x}{3}=\frac{2}{3}\\\\Simplify\\x=\frac{2}{3}[/tex]
Answer:
x = 2/3
Step-by-step explanation:
This equation can be solved by using conventional algebraic techniques.
We firstly want to make sure to get our constants on one side of the equation and then get our variables on the other.
6x - 10 = -8 + 3x Subtract 3x from both sides of the equation.
3x - 10 = -8 Add 10 to both sides of the equation.
3x = 2 Divide by 3 on both sides of the equation.
x = 2/3
Then, you can voluntarily check your answer.
6(2/3) - 10 = -8 + 3(2/3)
4 - 10 = -8 + 2
-6 = -6
HELP!!!!!!!!!! AP CALCULUS AB
Answers
Answer:
[tex]\lim_{x \to 5^{-}} f(x)= 54[/tex]
[tex]\lim_{x \to 5^{+}} f(x) = -17[/tex]
Step-by-step explanation:
+ means we are approaching from the right
- means we are approaching from the left
We are given that if x < 5, it is x² + 5x + 4
So to find [tex]\lim_{x \to 5^{-}} f(x)[/tex] we would plug in 5 into that piecewise function part:
5² + 5(5) + 4 = 54
We would be approaching y = 54 if we approach from the left.
We are given that x > 5, it is -4x + 3
So to find [tex]\lim_{x \to 5^{+}} f(x)[/tex] we would plug in 5 into that piecewise function part:
-4(5) + 3 = -17
We would approach y = -17 if we approached from the right.
Answer:
[tex]\lim_{x \to 5^-}f(x)=54\\\lim_{x \to 5^+}f(x)=-17[/tex]
Step-by-step explanation:
So, we have the function:
[tex]f(x)=x^2+5x+4,x<5\\f(x)=8,x=5\\f(x)=-4x+3,\text{ otherwise}[/tex]
And we want to show that this has a jump discontinuity.
As directed, calculate the limit at x=5 from the left and from the right. Thus:
[tex]\lim_{x \to 5^-}f(x)[/tex]
To calculate this, since we're coming from the left, x is less than 5. So, use the first equation:
[tex]\lim_{x \to 5^-}f(x)\\= \lim_{x \to 5^-}(x^2+5x+4)\\[/tex]
Use direct substitution:
[tex]=(5)^2+5(5)+4\\=25+25+4\\=54[/tex]
Thus, as the limit approaches f(5) from the left, the function approaches 54.
Now, calculate the limit at x=5 from the right. Since we're coming from the right, use the third equation:
[tex]\lim_{x \to 5^+}f(x)\\= \lim_{x \to 5^+}(-4x+3)[/tex]
Direct substitution:
[tex](-4(5)+3)\\=-20+3\\=-17[/tex]
Thus, as the limit approaches f(5) from the right, the function approaches -17.
As you can picture, there's a huge gap between y=54 and y=-17. The limits are not equal and both of the limits do exist. Thus, we have jump discontinuity.