![f(x,y)=2y^3+12x^2-24xy](https://tex.z-dn.net/?f=f%28x%2Cy%29%3D2y%5E3%2B12x%5E2-24xy)
Find the first derivatives:
![f'_x=24x-24y \\ f'_y=6y^2-24x](https://tex.z-dn.net/?f=f%27_x%3D24x-24y%20%5C%5C%20f%27_y%3D6y%5E2-24x)
.
Solve the system
![\left \{ {{f'_x=0} \atop {f'_y=0}} \right.](https://tex.z-dn.net/?f=%20%5Cleft%20%5C%7B%20%7B%7Bf%27_x%3D0%7D%20%5Catop%20%7Bf%27_y%3D0%7D%7D%20%5Cright.%20)
:
![\left \{ {{24x-24y=0} \atop {6y^2-24x=0}} \right. \rightarrow \left \{ {{x=y} \atop {6y^2-24y=0}}](https://tex.z-dn.net/?f=%20%5Cleft%20%5C%7B%20%7B%7B24x-24y%3D0%7D%20%5Catop%20%7B6y%5E2-24x%3D0%7D%7D%20%5Cright.%20%5Crightarrow%20%5Cleft%20%5C%7B%20%7B%7Bx%3Dy%7D%20%5Catop%20%7B6y%5E2-24y%3D0%7D%7D%20)
. The second equation has solutions
![y_1=0, \\ y_2=4](https://tex.z-dn.net/?f=y_1%3D0%2C%20%5C%5C%20y_2%3D4)
and then
![x_1=0, \\ x_2=4](https://tex.z-dn.net/?f=x_1%3D0%2C%20%5C%5C%20x_2%3D4)
and you have two points
![A_1(0,0), \\ A_2(4,4)](https://tex.z-dn.net/?f=A_1%280%2C0%29%2C%20%5C%5C%20A_2%284%2C4%29)
.
Find the first derivatives:
![f^{''}_{xx}=24 \\ f^{''}_{xy}=f^{''}_{yx}=-24 \\ f^{''}_{yy}=12y](https://tex.z-dn.net/?f=f%5E%7B%27%27%7D_%7Bxx%7D%3D24%20%5C%5C%20f%5E%7B%27%27%7D_%7Bxy%7D%3Df%5E%7B%27%27%7D_%7Byx%7D%3D-24%20%5C%5C%20f%5E%7B%27%27%7D_%7Byy%7D%3D12y)
and calculate
![\Delta=\left| \left[\begin{array}{cc}24&-24\\-24&12y\end{array}\right]\right |=24\cdot 12y-(-24)^2=288y-576](https://tex.z-dn.net/?f=%5CDelta%3D%5Cleft%7C%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D24%26-24%5C%5C-24%2612y%5Cend%7Barray%7D%5Cright%5D%5Cright%20%7C%3D24%5Ccdot%2012y-%28-24%29%5E2%3D288y-576)
.
Since
![\Delta(A_1)=-576](https://tex.z-dn.net/?f=%5CDelta%28A_1%29%3D-576%20)
and
![f^{''}_{xx}\ \textgreater \ 0](https://tex.z-dn.net/?f=f%5E%7B%27%27%7D_%7Bxx%7D%5C%20%5Ctextgreater%20%5C%200)
,
![A_1](https://tex.z-dn.net/?f=A_1)
is a point of maximum and
![f(0,0)=0](https://tex.z-dn.net/?f=f%280%2C0%29%3D0)
.
Since
![\Delta(A_2)=576](https://tex.z-dn.net/?f=%5CDelta%28A_2%29%3D576%20)
and
![f^{''}_{xx}\ \textgreater \ 0](https://tex.z-dn.net/?f=f%5E%7B%27%27%7D_%7Bxx%7D%5C%20%5Ctextgreater%20%5C%200)
,
![A_1](https://tex.z-dn.net/?f=A_1)
is a point of minimum and
![f(4,4)=-64](https://tex.z-dn.net/?f=f%284%2C4%29%3D-64)
.
Answer:
The first paragraph is (it explains that the wood frog is a unique animal
Second (explains how the wood tree frog can survive in cold weather
That’s all I know hope it helps
Answer:
Sienna has 5 3/4 pounds of grapes. She eats 1/8 pound and gives 2 1/2 pounds to her brother. How many pounds of grapes does Sienna have left?
Answer
5.0/5
2
The pounds of grapes left with seinna is 3.125 or pounds
Solution:
Given that Sienna has pounds of grapes
She eats pound and gives pounds to her brother
To find: pounds of grapes left with seinna
We can find the pounds of grapes left with seinna by subtracting the pounds of grapes she ate and pounds of grapes she gave to brother from the total pounds of grapes she had
Pounds of grapes left with seinna = total pounds of grapes - pounds of grapes she ate - pounds of grapes she gave to brother
Substituting the values we get,
Thus the pounds of grapes left with seinna is 3.125 or poundsStep-by-step explanation:
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Please post the picture if you forgot.