Find the minimum value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } - 4 y + 4$ on the hyperbola $x ^ { 2 } - y ^ { 2 } = 4$ .
A) $12 \sqrt { 3 }$
B) 43
C) 6
D) No minimum, unbounded below

Question

Quizplus · Accepted Answer

To find the minimum value of $f(x, y)$ on the hyperbola $x^2 - y^2 = 4$, substitute $y = \sqrt{x^2 - 4}$ into $f(x, y)$ and minimize the resulting function. The minimum value is 6.

Find the Minimum Value Of f(x,y)=x2+y2−4y+4f ( x , y ) = x ^ { 2 } + y ^ { 2 } - 4 y + 4f(x,y)=x2+y2−4y+4

Find the Minimum Value Of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } - 4 y + 4$