X = 0 y =-1 because you have to have to find numbers to replace x & y.
Answer:
A. 
Step-by-step explanation:
A. The problems asked for 2 ways to solve it, expanding the equation with the substitution x(t)=2 cos(t) and y(t)=4 sin(t) to differentiate it . The other way is by chain rule.
Expanding and differentiating:
We start by substituting x(t)=2 cos(t) and y(t)=4 sin(t) in h(x,y)=4x2+y2:
So, in the path that the hiker chose:
Chain rule:
We start differentiating h(x,y) using chain rule as follows:
Now, it´s easy to find all these derivatives:
Now we replace them in the chain rule, with the replacement x=2cos(t) and y=4sin(t) in the x,y that are left and we operate everything:
This will be our answer
X4)-(2•(x3)))-32x2)+18x
STEP
2
:
Equation at the end of step
2
:
(((x4) - 2x3) - 32x2) + 18x
STEP
3
:
STEP
4
:
Pulling out like terms
4.1 Pull out like factors :
x4 - 2x3 - 9x2 + 18x =
x • (x3 - 2x2 - 9x + 18)
Checking for a perfect cube :
4.2 x3 - 2x2 - 9x + 18 is not a perfect cube
Trying to factor by pulling out :
4.3 Factoring: x3 - 2x2 - 9x + 18
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -9x + 18
Group 2: x3 - 2x2
Pull out from each group separately :
Group 1: (x - 2) • (-9)
Group 2: (x - 2) • (x2)
-------------------
Add up the two groups :
(x - 2) • (x2 - 9)
Which is the desired factorization
Trying to factor as a Difference of Squares:
4.4 Factoring: x2 - 9
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 9 is the square of 3
Check : x2 is the square of x1
Factorization is : (x + 3) • (x - 3)
Final result :
x • (x + 3) • (x - 3) • (x - 2)
