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The shaded area includes everything ABOVE the line ...but NOT the line

equation of the line is y = 1/3x -3

y > 1/3x -3

8 minutes / (5x400)m = .004 min / meter

.004 min/meter * (9 * 200 meters) = 7.2 mins

This is a partition problem of 14 into 5 parts = 23. Ordinarily, order does not matter in partitions. But, if it is, then just multiply by 2 and REVERSE the order = 23 x 2 = 46.

10 + 1 + 1 + 1 + 1 = 14
9 + 2 + 1 + 1 + 1 = 14
8 + 3 + 1 + 1 + 1 = 14
8 + 2 + 2 + 1 + 1 = 14
7 + 4 + 1 + 1 + 1 = 14
7 + 3 + 2 + 1 + 1 = 14
7 + 2 + 2 + 2 + 1 = 14
6 + 5 + 1 + 1 + 1 = 14
6 + 4 + 2 + 1 + 1 = 14
6 + 3 + 3 + 1 + 1 = 14
6 + 3 + 2 + 2 + 1 = 14
6 + 2 + 2 + 2 + 2 = 14
5 + 5 + 2 + 1 + 1 = 14
5 + 4 + 3 + 1 + 1 = 14
5 + 4 + 2 + 2 + 1 = 14
5 + 3 + 3 + 2 + 1 = 14
5 + 3 + 2 + 2 + 2 = 14
4 + 4 + 4 + 1 + 1 = 14
4 + 4 + 3 + 2 + 1 = 14
4 + 4 + 2 + 2 + 2 = 14
4 + 3 + 3 + 3 + 1 = 14
4 + 3 + 3 + 2 + 2 = 14
3 + 3 + 3 + 3 + 2 = 14

Factor completely: x^8 - 12x^4 - 64.

Hello Guest!

\(x^8 - 12x^4 - 64 \)

\(u=x^4\)

\(u^2-12u-64=0\)

           p         q

\(u=-\frac{p}{2}\pm\sqrt{(\frac{p}{2})^2-q}\\ u=6\pm\sqrt{36+64}\\ u=6\pm 10\)

\(u_{1,2}=16\\ u_{3,4}=-4 \)

\(u=x^4\)

\(x^8 - 12x^4 - 64 \)

\(=(x^4-16)(x^4+4)\\ =(x^2+4)(x+2)(x-2)(x^4+4)\)

  !

Guest is probably right but I do not think you can really tell just by looking at this graph.

I am not familiar with the terms cusps and corners.

It looks like a piecemeal graph.  So its ability to be differentiable at the joining points would depend on exactly what the pieces are.

When is a graph differentiable, the definition from Google:

"A function is differentiable at a point when there's a defined derivative at that point. This means that the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right."

Using CRT and MMI

C - 3 mod 7 = 0,

C - 6 mod 8 = 0, 

C+3  mod 5 = 0, solve for C

C =280n + 262, where n =0, 1, 2, 3.......etc.

The smallest number of candies in the box = 262

9.81 =[6.67428E-11 * 1 * M] / (6371000)^2, solve for M

M = 5.966 x 10^24 kg.

Hello, Guest! Did you solve this "difficult" problem? 

Here is the completed Venn..... any questions?

*** edit ***    that 44 should be a   ' 3'

A rate = 100 l  / 5 min =   20 l/min

B rate = 100/4 = 25 l/min                        combined = 45 liters per minute

9000 l  /  45 l/ min = 200 min

this is hard

Then your question is not correct!!!

edge * edge * height / 3 = volume

1st answer is not correct but the 2nd is.

If   xy  =  - 4x       

        the equation devolves to something other than one  = 1

     xy = - 4x

      y = -4

7x+20      +     9x-4    =    180 ^o  

The possible values of A are -5 and -6.

The possible values of y are -1 and -3.

The equation of the line is 2x - y - 2 = 0.

mmi = Modular Multiplicative Inverse of 1 to 4! mod 4!.

1  -  The mmi =  1
5  -  The mmi =  5
7  -  The mmi =  7
11  -  The mmi =  11
13  -  The mmi =  13
17  -  The mmi =  17
19  -  The mmi =  19
23  -  The mmi =  23

m =1 x 5 x 7 x 11 x 13 x 17 x 19 x 23 =37,182,145
37,182,145 mod 4! = 1

No, the area is 9 units squared.

Legs are (approximately)  2.35425  and  7.64575

Ey i found it its y=-4 :D

The possible values of y are -1 and -2.

Are you in aops 

If you select the Insert tab in MS Word then you should see Equation and Symbol options.

f(x) is differentiable everywhere

there are no cusps or corners -- no sharp turns/curves anywhere on the graph

(cusp on the left, corner on the right)

no vertical tangents anywhere, there's a horizontal tangent though at x=0

there are also no discontinutities since the function is continuous everywhere