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
What does 4(2n+10)=2n+4n+10 equal?
Dmitry_Shevchenko [17]

Answer:

n= -15 i belive thats right

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
a student answers 90% of the questions on a math exam correctly. if he answers 27 questions correctly how many questions are on
kodGreya [7K]

Answer:

30

Step-by-step explanation:

6 0
3 years ago
How are the variables on the graph related?
Anna007 [38]
Having a look at the graph, we will find that:
For speed: it is increasing along the x-axis
For height: it is increasing along the y-axis

This means that the function is linearly increasing.

Based on this, the correct choice would be:
D. As speed increases, height increases
8 0
3 years ago
Is 2/13a repeating decimal
Veseljchak [2.6K]

Answer:

No

Step-by-step explanation:

2/13=0.15384615

6 0
3 years ago
Quick question
zaharov [31]

Answer:

False

Step-by-step explanation:

Let us evaluate the problem using the proper order of operation:

           4x − 2(5x − 1)

To solve this problem, we use the order of operation called PEMDAS;

P - Parentheses

E - Exponents

M - Multiplication

D - Division

A - Addition

S - Subtraction

Now, since operations inside the parentheses are of the utmost priority, let us solve this;

             4x − 2(5x − 1)

            = 4x - 10x +2

            = -6x + 2  or 2 - 6x

We see that the solution give is false.

7 0
3 years ago
Read 2 more answers
Other questions:
  • O graph the equation 2x + 5y = 10, Zeplyn draws a line through the points (5, 0) and (0, 2). What is the slope of the line repre
    13·2 answers
  • The lower base of the frustum shown in the figure is a square 8ft on each side. The upper base is a square 4ft on each side. In
    10·1 answer
  • Simplify sqrt of 4x^6
    9·1 answer
  • If the monthly cost for 23 HCF is <br> $48.31 , what is the monthly cost for 18 HCF?
    6·1 answer
  • Point B is on line segment AC. Given AB = 2x + 10, BC = x, and AC = 5x,
    7·2 answers
  • How many terms are in the algebraic expression -7 + 12x4 - 5y8 + x?
    10·1 answer
  • Is segment ST tangent to circle P1
    6·1 answer
  • Determine the y-intercept of the graph of this equation: y - 3 = 2(x + 5)
    10·1 answer
  • -4b + 8c + 12 - 8b - 2c + 6. what is the value of the expression when b=2 and c= -3? Enter your answer.
    11·1 answer
  • What number is represented in the expanded notation below?
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!