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
I need help fast !!
AysviL [449]
<h2>Answer:</h2>

Luke does a work of 308Nm

<h2>Step-by-step explanation:</h2>

If we have a constant force \vec{F} that acts on a body in the same direction as the displacement \vec{s}, then the work W is defined as the product of the force magnitude F and the displacement magnitude s. In other words:

W=Fs, which is valid for a constant force in direction of straight-line displacement.

In this problem:

F=22N \\ \\ s=14m

Therefore:

W=22\times 14 \\ \\ \boxed{W=308Nm}

3 0
3 years ago
PLEASE HELPPPP I NEED THIS FASTTTTT
rusak2 [61]

Answer:

there \: are \: 8 \: values \\  =  \frac{8}{17}  \times 100\% \\  = 47\%

6 0
2 years ago
Shay mal e 8 2/3 cups of snacks mix she puts all the snack mix into plastic bags she puts 1/3 cup of the snack mix in each bag h
AnnyKZ [126]

Answer:

She needs 26 plastic bags.

Step-by-step explanation:

Given:

Amount of cups of snacks mix = 8 2/3

Rewriting 8 2/3 we get,

Amount of snacks mix = \frac{26}{3}

Number of Cups in each bag = \frac{1}{3}

She puts all the snacks mix in plastic bags

We need to find the number of plastic bags required to put all the snacks mix

Let number of plastic bag be x.

hence we can say that Number of Cups in each bag multiplied by number of plastic bag is equal to  Amount of snacks mix.

\frac{1}{3}x=\frac{26}{3}\\\\x= \frac{26\times3}{3\times1} =26

Hence total number of plastic bags she needs is 26.

6 0
3 years ago
Is {(5, 12), (-4,9), (-2,-7), (-4,0), (3, 2)}<br> A function? Please help me
Anna71 [15]

Answer:

Not a function

Step-by-step explanation:

(-4,9), (-4,0)   : one input (-4) with 2 output -9 , 0

6 0
3 years ago
Find the sum of 7a^3 + 14a +12 and -6a^3 +12a^2 -7
SOVA2 [1]
To find the sum:

{7a}^{3} + 14a + 12  + ( {- 6a }^{3} ) + {12a }^{2}   - 7 \\  = {7a }^{3}   -  {6a}^{3}  + {12a }^{2}  + 14a + 12 - 7 \\  =  {a}^{3}  + {12a }^{2}  + 14a + 5
Therefore the answer is a^3+12a^2+14a+5.
Hope it helps!
8 0
3 years ago
Read 2 more answers
Other questions:
  • 5 − (x + 5) &gt; −2(x + 4)
    7·2 answers
  • PLZ HELP! I need to turn this in urgently :))
    13·2 answers
  • (0, -5), (1, 0), (2, 5), (3, 10)<br> in ordered pairs
    13·1 answer
  • Alan invests some of his retirement savings in a fund earning 4% simple interest and invests twice as much in a fund earning 6%
    7·1 answer
  • What is the side length of a square with a perimeter of 64 meters
    15·1 answer
  • al doilea termen al unei adunari este de 6 ori mai mare decat primul.Daca as aduna primului numar 4,acesta ar deveni treimea cel
    12·1 answer
  • Please help me! i dont understanddd! i have no clue how to do this unit :(
    9·1 answer
  • 2+3×=2×+3 ---&gt; ×=1<br><br><br> Help please
    6·1 answer
  • 20 POINTS<br><br> Part E<br> How many of the 320 million Americans would you predict wear glasses?
    15·1 answer
  • Someone pls help! 3x ^2 and x = 10 pls help!
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!