minimum value of an expression?

Question

whats the minimum value of the expression 2x^2+3y^2-4*x-12*y+18 ?

1)18
2)10
3)4
4)0

>2(x - 1)^2 + 3(y - 2)^2 = ? - 4
>? can't be less that 4


how ?

Answer 1

2x^2 + 3y^2 - 4*x - 12*y + 18 = ?
2(x^2 - 2x + 1) +3(y^2 - 4y + 4) + 18 = ? + 2 + 12
2(x - 1)^2 + 3(y - 2)^2 + 18 = ? + 14
2(x - 1)^2 + 3(y - 2)^2 = ? - 4
At this point you can see that this is an ellipse.
2(x - 1)^2 + 3(y - 2)^2 = ? - 4
? can't be less that 4

4 is the minimum.

Answer 2

enable u = x*sinx i'm attempting to stay with your parenthesis right here. The function reduces to f(u) = 216/5 u^2 + 36u + 30 + 25/u + one hundred twenty five/6u^2 Differentiating as quickly as and rearranging u^3 * f'(u) = 432/5 * u^4 + 36u^3 - 25u - one hundred twenty five/3 The quartic polynomial on the RHS has 2 rational roots at u = +/- 5/6 f(5/6) = a hundred and fifty f(-5/6) = 0 an ordinary plot exhibits that f(5/6) = a hundred and fifty is the minimum we seek for. we are in a position to even bypass directly to instruct that f"(5/6) > 0, yet that may no longer needed. ANS: a hundred and fifty there'll actual be 2 minima for the duration of this variety because xsinx = 5/6 has 2 recommendations. The graph looks something like a W. *EDIT* Phineas, I have been given one hundred twenty in the commencing up by using fact I forgot the consistent 30 inthe unique function. yet I corrected that mistake.

Answer 3

I think its 4