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
anyanavicka [17]
3 years ago
7

Consider the equation below. f(x) = 2x^3 + 3x^2 − 12x (a) Find the interval on which f is increasing. (Enter your answer in inte

rval notation.) Find the interval on which f is decreasing. (Enter your answer in interval notation.) (b) Find the local minimum and maximum values of f. local minimum local maximum (c) Find the inflection point. (x, y) = Find the interval on which f is concave up. (Enter your answer in interval notation.) Find the interval on which f is concave down. (Enter your answer in interval notation.)

Mathematics
1 answer:
gogolik [260]3 years ago
4 0

Answer:

a) increasing: (-∞, -2)∪(1, ∞); decreasing: (-2, 1)

b) local maximum: (-2, 20); local minimum: (1, -7)

c) inflection point: (-0.5, 6.5); concave up: (-0.5, ∞); concave down: (-∞, 0.5)

Step-by-step explanation:

A graphing calculator can show you the local extremes. Everything else falls out from those.

a) The leading coefficient is positive, so the general shape of the graph is from lower left to upper right. The function will be increasing from -infinity to the local maximum (x=-2), decreasing from there to the local minimum (x=1), then increasing again to infinity.

__

b) See the attached. This is what we did first. If you want to do this by hand, you find where the derivative is zero:

  6x^2 +6x -12 = 0

  6(x+2)(x-1) = 0 . . . . . local maximum at x=-2; local minimum at x=1.

You know the left-most zero of the derivative is the local maximum because of the nature of the curve (increasing, then decreasing, then increasing again).

The function values at those points are easily found by evaluating the function written in Horner form:

  ((2x +3)x -12)x

at x=-2, this is ((2(-2)+3)(-2) -12)(-2) = (2-12)(-2) = 20 . . . . (-2, 20)

at x = 1, this is 2 +3 -12 = -7 . . . . . . . . . . . . . . . . . . . . . . . . . (1, -7)

__

c) The point of inflection of a cubic is the midpoint between the local extremes: ((-2, 20) +(1, -7))/2 = (-1, 13)/2 = (-0.5, 6.5)

A cubic curve is symmetrical about the point of inflection. When you consider the derivative is a parabola symmetric about the vertical line through its vertex, perhaps you can see why. The local extremes of the cubic are the zeros of the parabola, which are symmetric about that line of symmetry. Of course the vertex of the derivative (parabola) is the place where its slope is zero, hence the second derivative of the cubic is zero--the point of inflection.

__

The cubic is concave down to the left of the point of inflection; concave up to the right of that point. The interval of downward concavity corresponds to the interval on which the first derivative (parabola) is decreasing or the second derivative is negative.

You might be interested in
What is one way that authors can come up with ideas for providing context for their readers?(1 point)
nataly862011 [7]

Answer:

Brainstorming i think

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
Given g(x) = -x – 5, find g(5).
dimulka [17.4K]
Answer is -10

Work: -(5+5)
8 0
3 years ago
Estimate the square root of 75.<br> a. 8.7<br> b. 4.2<br> c. 7.4<br> d. 9.5
SVEN [57.7K]

Answer: A.8.7

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
Rates and ratios simalar
Alex777 [14]
Rate is closer to a constant pattern , example: I'm walking at a rate of 2 miles an hour.

Ratio is related to the proportion of two or more objects, example: The ratio for a perfect tang juice is 1 packet of tang dust thingy for 1.5 liters of water.

Hope it helps :)
4 0
2 years ago
Justin has a savings account.
andrey2020 [161]

Answer:

220

Step-by-step explanation:

Multiply 15 times 10 =150

150+70=220

Hope this helps :)

6 0
2 years ago
Read 2 more answers
Other questions:
  • Kimmy has played video games 24 hours over the last 9 days. She played either 2 hours a day or 3 hours a day. During the 9 day p
    8·2 answers
  • If you flip three fair coins, what is the probability that you'll get two tailed and one head in any order
    5·2 answers
  • A paper company needs to ship paper to a large printing business. The paper will be shipped in small boxes and large boxes. The
    12·1 answer
  • Plz Help I would appreciate it!!!
    6·1 answer
  • Enter your answer and show all the steps that you use to solve this problem in the space provided.
    7·1 answer
  • My number rounded to the nearest tenth is 3.6. What might my number be? Justify your answer.
    6·1 answer
  • Helpp me hw is so stressing
    13·1 answer
  • Gabe paid $ 120 for 3 shirts and 2 pairs of pants.
    12·1 answer
  • Please can someone help with this question. I don't understand
    6·1 answer
  • Which expresion is equivilenet to 3/24
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!