CALCULUS help!!!!?

Question

I am trying to help my kid with these problems - I do not have time to read the book. please help me to help her understand.

y= 10a + 2b + 10ab - 10a2 - 5b2 (10a squared; 5b squared)

Estimate the value if (a) and (b) when (y) is maxmized.

all help will be greatly appreciated

Answer 1

start by taking the differential of y with respect to a
treating b as a constant
dy/da=10+10b-20a
then the differential dy/db treating a as a constant
dy/db=2+10a-10b

then set the differentials to xero and solve the simultaneous equations for a and b:

0=10+10b-20a
0=2-10b+10a

add the equations

0=12-10a
a=6/5

b=7/5

I graphed this to make sure the surface was maximized in three dimensional space at this point. It came out correctly.

Here's the data:
a b y
-0.2 0 -2.4
0 0.2 0.2
0.2 0.4 2.4
0.4 0.6 4.2
0.6 0.8 5.6
0.8 1 6.6
1 1.2 7.2
1.2 1.4 7.4
1.4 1.6 7.2
1.6 1.8 6.6
1.8 2 5.6
2 2.2 4.2
2.2 2.4 2.4
2.4 2.6 0.2
2.6 2.8 -2.4

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