1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
mrs_skeptik [129]
3 years ago
5

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

asing. (Enter your answer using interval notation.) Find the interval on which f is decreasing. (Enter your answer using interval notation.) (b) Find the local minimum and maximum values of f. local minimum value local maximum value (c) Find the inflection point. (x, y) = Find the interval on which f is concave up. (Enter your answer using interval notation.) Find the interval on which f is concave down. (Enter your answer using interval notation.)
Mathematics
1 answer:
Ainat [17]3 years ago
4 0

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>

You might be interested in
TIME REMAINING
bezimeni [28]
D. city A is more likely because they have the highest population density just divide the population by number of miles that the city covers and that will show you your density
4 0
2 years ago
Round 835.411 to the nearest tenth.
kondor19780726 [428]

The tenth position is the one that goes after the decimal point.

To round a number, you have to take into account the following:

1. If the number that goes after the position we are going to round to is greater than 5, we round to the next number in that position.

2. If the number that goes after the position we are going to round to is less than 5, we round to the same number in that position.

In this case, the number that is on the tenth's position is 4. The number that is after this position is 1, which is less than 5, then we round the number in this positon to 4.

The rounded number would be:

835.4

4 0
1 year ago
The length of 2 sides on an isosceles triangle is 15. What is the range of possible lengths for the other leg?
erica [24]

Answer:

34

Step-by-step explanation:

4 0
3 years ago
F(x)=16x^4-81=0 select all zeros of the function !!!
atroni [7]

Answer:

3/2, -3/2, (if it is asking for imaginary solutions as well then 3i/2 and -3i/2)

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
For a Cinco de Mayo party, Alan is placing sets of chips and salsa's throughput the house. He has 20 bags of chips and 15 jars o
NemiM [27]
HCF. The answer is 5
7 0
2 years ago
Read 2 more answers
Other questions:
  • 88.83 is what percent of 21?
    9·2 answers
  • How can you use 2x8 to find 4x8
    12·2 answers
  • Steven’s mother asked him to buy 750 grams of sugar. At the supermarket, he finds that the sugar packages give the weight in kil
    11·1 answer
  • A supermarket has found that 60% of their customers spend more than $75 each visit. Whats the probability that the next 2 custom
    6·1 answer
  • For the hypothesis test H0: μ = 10 against H1: μ &gt;10 and variance known, calculate the Pvalue for each of the following test
    15·1 answer
  • Paul went to a movie and had 200$ to spend on himself and his friends.If a movie ticket costs 8$ and he wants to buy 40$ worth o
    11·1 answer
  • CANT seem to find the answer
    11·2 answers
  • use the discrminant to determine all values of k which would result in the equation -3x^2-6x+k=0 having real,unequal roots.​ mus
    14·1 answer
  • Buddy guy!!, has anyone noticed that mos thanks have bats literally lots of there names r like dome7
    5·1 answer
  • Graph the following state and show the a h and k values. <br> y=2(x+3)^2-4
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!