By definition of absolute value,
![g(x)=6x|x|=\begin{cases}6x^2&\text{for }x\ge0\\-6x^2&\text{for }x](https://tex.z-dn.net/?f=g%28x%29%3D6x%7Cx%7C%3D%5Cbegin%7Bcases%7D6x%5E2%26%5Ctext%7Bfor%20%7Dx%5Cge0%5C%5C-6x%5E2%26%5Ctext%7Bfor%20%7Dx%3C0%5Cend%7Bcases%7D)
Differentiating once gives
![g'(x)=\begin{cases}12x&\text{for }x>0\\?&\text{for }x=0\\-12x&\text{for }x](https://tex.z-dn.net/?f=g%27%28x%29%3D%5Cbegin%7Bcases%7D12x%26%5Ctext%7Bfor%20%7Dx%3E0%5C%5C%3F%26%5Ctext%7Bfor%20%7Dx%3D0%5C%5C-12x%26%5Ctext%7Bfor%20%7Dx%3C0%5Cend%7Bcases%7D)
As
![x\to0](https://tex.z-dn.net/?f=x%5Cto0)
from either side, we find that
![g'(x)\to0](https://tex.z-dn.net/?f=g%27%28x%29%5Cto0)
, so really
![g'(x)=\begin{cases}12x&\text{for }x\ge0\\-12x&\text{for }x](https://tex.z-dn.net/?f=g%27%28x%29%3D%5Cbegin%7Bcases%7D12x%26%5Ctext%7Bfor%20%7Dx%5Cge0%5C%5C-12x%26%5Ctext%7Bfor%20%7Dx%3C0%5Cend%7Bcases%7D)
Differentiating again, we find
![g''(x)=\begin{cases}12&\text{for }x>0\\?&\text{for }x=0\\-12&\text{for }x](https://tex.z-dn.net/?f=g%27%27%28x%29%3D%5Cbegin%7Bcases%7D12%26%5Ctext%7Bfor%20%7Dx%3E0%5C%5C%3F%26%5Ctext%7Bfor%20%7Dx%3D0%5C%5C-12%26%5Ctext%7Bfor%20%7Dx%3C0%5Cend%7Bcases%7D)
but this time, the limit does not exist as
![x\to0](https://tex.z-dn.net/?f=x%5Cto0)
from either side, so
![g''(0)](https://tex.z-dn.net/?f=g%27%27%280%29)
is undefined. However, we see that
![g''(x)>0](https://tex.z-dn.net/?f=g%27%27%28x%29%3E0)
when
![x>0](https://tex.z-dn.net/?f=x%3E0)
, and
![g''(x)](https://tex.z-dn.net/?f=g%27%27%28x%29%3C0)
when
![x](https://tex.z-dn.net/?f=x%3C0)
, and we know that
![g(x)](https://tex.z-dn.net/?f=g%28x%29)
is continuous at
![x=0](https://tex.z-dn.net/?f=x%3D0)
. This means the concavity must change at
![x=0](https://tex.z-dn.net/?f=x%3D0)
, so (0, 0) is the inflection point. (The takeaway is that inflection points *can* occur when the second derivative is undefined, but not always.)
Remark
Yes even though 8 is not a prime number. There is nothing that will divide into 3 or 8 the will give a whole number answer when the division is complete.
That means that 3 and 8 are prime to each other in base 10.
Answer:
3, 8, 12, 12, 14, 20, 21, 23, 26, 34
stem leaf
0 3 8
1 2 2 4 (you have 12,12,14)
2 0 1 3 6 ( numbers 20,21,23,26)
3 4 ( number 34)
Answer:
p = 1.2
Step-by-step explanation:
-10.9p = -9.18 - 3.9
-10.9p = -13.08
Divide both sides by -10.9
p = -13.08/-10.9
p = 1.2
Answer:
3/10, 5/8, 11/12
Step-by-step explanation: