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larisa [96]
2 years ago
11

I need help with problem 1 with a through explanation and solution please ​

Mathematics
1 answer:
ki77a [65]2 years ago
6 0

Explanation:

The cubic ...

  f(x) = ax³ +bx² +cx +d

has derivatives ...

  f'(x) = 3ax² +2bx +c

  f''(x) = 6ax +2b

<h3>a)</h3>

By definition, there will be a point of inflection where the second derivative is zero (changes sign). The second derivative is a linear equation in x, so can only have one zero. Since it is given that a≠0, we are assured that the line described by f''(x) will cross the x-axis at ...

  f''(x) = 0 = 6ax +2b   ⇒   x = -b/(3a)

The single point of inflection is at x = -b/(3a).

__

<h3>b)</h3>

The cubic will have a local extreme where the first derivative is zero and the second derivative is not zero. These will only occur when the discriminant of the first derivative quadratic is positive. Their location can be found by applying the quadratic formula to the first derivative.

  x=\dfrac{-2b\pm\sqrt{(2b)^2-4(3a)(c)}}{2(3a)} = \dfrac{-2b\pm\sqrt{4b^2-12ac}}{6a}\\\\x=\dfrac{-b\pm\sqrt{b^2-3ac}}{3a}\qquad\text{extreme point locations when $b^2>3ac$}

There will be zero or two local extremes. A local extreme cannot occur at the point of inflection, which is where the formula would tell you it is when there is only one.

__

<h3>c)</h3>

Part A tells you the point of inflection is at x= -b/(3a).

Part B tells you the midpoint of the local extremes is x = -b/(3a). (This is half the sum of the x-values of the extreme points.) You will notice these are the same point.

The extreme points are located symmetrically about their midpoint, so are located symmetrically about the point of inflection.

_____

Additional comment

There are other interesting features of cubics with two local extremes. The points where the horizontal tangents meet the graph, together with the point of inflection, have equally-spaced x-coordinates. The point of inflection is the midpoint, both horizontally and vertically, between the local extreme points.

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Are the equations 9x=5x+4 and 14x=4 equivilant?​
Paha777 [63]

Answer:

No

Step-by-step explanation:

First solve both equations:

1) 9x = 5x + 4

Simplifying

9x = 5x + 4

Reorder the terms:

9x = 4 + 5x

Solving

9x = 4 + 5x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-5x' to each side of the equation.

9x + -5x = 4 + 5x + -5x

Combine like terms: 9x + -5x = 4x

4x = 4 + 5x + -5x

Combine like terms: 5x + -5x = 0

4x = 4 + 0

4x = 4

Divide each side by '4'.

x = 1

Simplifying

x = 1

2) 14x = 4

14x = 4 (divide both sides by 14 to get x)

14x/14 = 4/14

x = 0.285714285714

As you can see, the value of x in the second equations is less than one, therefore making these algebraic equations not equivalent.

4 0
2 years ago
Read 2 more answers
X-3y=1<br> -x+6y=-7<br><br> Solve the system using elimination <br> Type ordered pair
Vedmedyk [2.9K]

Answer:

x=-5, y=-2. (-5, -2).

Step-by-step explanation:

x-3y=1

-x+6y=-7

----------------

3y=-6

y=-6/3

y=-2

x-3(-2)=1

x+6=1

x=1-6

x=-5

3 0
3 years ago
What is -8?<br> PLEASE HELP ME
Mrac [35]

-8 IS A NEGATIVE INTEGER

6 0
3 years ago
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Use pi=3.14 to estimate the circumference of the circle to the nearest hundreth the diameter is 11 cm
TiliK225 [7]
The formula of circumference: 兀×diameter

In this case:

3.14×11
=34.54cm

Nearest hundreth = 2 demical places

Therefore, the answer is 34.54cm.

Hope it helps!
3 0
2 years ago
Maximim height of the diver
Mila [183]

Answer:

16

Step-by-step explanation:

7 0
2 years ago
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