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
Vikentia [17]
3 years ago
13

F(x) = x2 − x − ln(x) (a) find the interval on which f is increasing. (enter your answer using interval notation.) find the inte

rval on which f is decreasing. (enter your answer using interval notation.) (b) find the local minimum and maximum value 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:
Mekhanik [1.2K]3 years ago
6 0

Answer:

(a) Decreasing on (0, 1) and increasing on (1, ∞)

(b) Local minimum at (1, 0)

(c) No inflection point; concave up on (0, ∞)

Step-by-step explanation:

ƒ(x) = x² - x – lnx

(a) Intervals in which ƒ(x) is increasing and decreasing.

Step 1. Find the zeros of the first derivative of the function

ƒ'(x) = 2x – 1 - 1/x = 0

           2x² - x  -1 = 0

     ( x - 1) (2x + 1) = 0

         x = 1 or x = -½

We reject the negative root, because the argument of lnx cannot be negative.

There is one zero at (1, 0). This is your critical point.

Step 2. Apply the first derivative test.

Test all intervals to the left and to the right of the critical value to determine if the derivative is positive or negative.

(1) x = ½

ƒ'(½) = 2(½) - 1 - 1/(½) = 1 - 1 - 2 = -1

ƒ'(x) < 0 so the function is decreasing on (0, 1).

(2) x = 2

ƒ'(0) = 2(2) -1 – 1/2 = 4 - 1 – ½  = ⁵/₂

ƒ'(x) > 0 so the function is increasing on (1, ∞).

(b) Local extremum

ƒ(x) is decreasing when x < 1 and increasing when x >1.

Thus, (1, 0) is a local minimum, and ƒ(x) = 0 when x = 1.

(c) Inflection point

(1) Set the second derivative equal to zero

ƒ''(x) = 2 + 2/x² = 0

             x² + 2 = 0

                   x² = -2

There is no inflection point.

(2). Concavity

Apply the second derivative test on either side of the extremum.

\begin{array}{lccc}\text{Test} & x < 1 & x = 1 & x > 1\\\text{Sign of f''} & + & 0 & +\\\text{Concavity} & \text{up} & &\text{up}\\\end{array}

The function is concave up on (0, ∞).

You might be interested in
3p + 2q = 14<br> 10p + 6q = 44<br> What is p and what is q
Airida [17]

Answer:

p = 2 ; q = 4

Step-by-step explanation:

Given tbe equation :

3p + 2q = 14 - - - (1)

10p + 6q = 44 - - -(2)

What is p and what is q

This is a simultaneous equation ; using elimination method :

Multiply (1) by 6 and (2) by 2

18p + 12q = 84 - - - - (3)

20p + 12q = 88 - - - (4)

Subtract (3) and (4)

-2p = - 4

p = 4/2

p = 2

Put p = 2 in (1)

3p + 2q = 14

3(2) + 2q = 14

6 + 2q = 14

2q = 14 - 6

2q = 8

q = 8/2

q = 4

p = 2 ; q = 4

4 0
3 years ago
The sales tax for an item was $15.60 and it cost $390 before tax. Find the sales tax rate. Write your answer as a percentage.
labwork [276]

Answer:

15.60 times 100 divided by 390=4%

Step-by-step explanation:

7 0
3 years ago
Is 5/4 greater than , less than , or equal to the square root of 2
victus00 [196]

Answer:

less than

Step-by-step explanation:

the square root of 2 is 4 and 5/4 is 1.25

8 0
3 years ago
Two skateboarders start a race at the same time. Skateboarder A travels at a steady rate of 15 feet per second. Skateboarder B t
notsponge [240]
So covert 4 minutes to seconds. Which would be 4 times 60 which would mean there would be 240 seconds. So you would multiply 15 x 240 = 3,600 feet. Then multiply 22 x 240 = 5,280 feet. 5,280 - 3,600 = 1,680 feet.

ANSWER:
Skateboarder B will have traveled 1,680 more feet over 4 minutes.
5 0
3 years ago
This figure represents a ramp for a dollhouse that is being covered with fabric on all sides. How much fabric is needed to cover
astra-53 [7]

Answer: 144 cm^2

Step-by-step explanation:

i did the quiz

4 0
2 years ago
Read 2 more answers
Other questions:
  • There 6 pounds of apples in one bag. Make a table to show how many pound of apples are in 6 bags. How do you your answer is easo
    6·2 answers
  • The school that molly goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 6 sen
    15·1 answer
  • What kind of questions should you ask during an interview
    14·1 answer
  • Need help solving this
    10·1 answer
  • ASAP!! HELP!!!
    12·1 answer
  • jack was planting a tree. he was to dig a hole that was 3.25 feet deep for every 5.7 feet of tree height . how deep should he di
    6·1 answer
  • Question 3 of 10
    14·1 answer
  • At time t= 0, Elevator A is 40 feet above the ground floor.
    13·1 answer
  • Help me find p.....​
    11·1 answer
  • A submarine that is already at 605 feet below sea level ascends (rises up) 475 feet before descending (sinking down) another 255
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!