Test tomorrow! Help <3

[Diavdhdd222]

Hello, I have a maths test tomorrow, and the class has been given a handful of tasks prior to the test.

One of them involving the following:
R(x,y) = ax + 3y. What must R(x,y) be so that R(x,y) has the biggest value of 60 in the point C.
Here's a figure of the graph:


Another task I need help with is:
K(x)=0,25x^2+60x+5000 x is an element of [50, 300]
The company sells each product for 140, and the company manages to sell their stock out.
a) When is the production in profit?
b) Find the biggest profit for the company

Another model f for the costs is f(x)=0,25x^2+px+5000, where x is an element of [50, 300]
If the company decides on using this model, the profit is biggest when they produce and sell 200 items a week.
c) Solve the value of p.



(LINEAR OPTIMIZATION)
A small company has created two types of phone covers:
The company sells x covers of the type Filipp and y types of Compact.
The inequalities below contain the capacity constraints in the production:

a) Mark the graph area in which all of the inequalities meet one another.

The company sells Filipp for 150 dollars a piece and 225 dollars of Compact a piece.
b) How many covers need to be produced and sold of each type to obtain biggest possible income?
What's the biggest income the company can get a day?

Also:
Derivative:
f(x)=x^3+3x^2-4
The graph has a tangent with the slope 9. In the equation for the tangent, the constant term is negative.
Find the equation for the tangent.

FOR THE FIRST AND FINAL TASK, I CANNOT USE TOOLS, BUT FOR THE REST, I AM ALLOWED TO.

 

[tkhunny]

Please show your work.

For this, R(x,y) = ax + 3y, you have a family of lines... For R(x,y) = 0, we see y = -(a/3)x.

Thus, if it passes through C(6,2), we have 2 = -(a/3)*6 = -2a ==> a = -1
Finally, R(6,2) = -6 + 3*2 = 0, which is what we suspected.

Do you see how you might go about it by simply assuming R(x,y) = 60?