Find the minimum and maximum value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ on the ellipse $3 x ^ { 2 } + 2 y ^ { 2 } = 1$ .
A) Minimum $\frac { 1 } { 4 }$ , maximum $\frac { 5 } { 12 }$
B) Minimum $\frac { 1 } { 3 }$ , maximum $\frac { 1 } { 2 }$
C) Minimum $\frac { 1 } { 6 }$ , maximum $\frac { 5 } { 6 }$
D) No maximum or minimum, unbounded

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Quizplus · Accepted Answer

To find the minimum and maximum of $f(x, y) = x^2 + y^2$ subject to the constraint $3x^2 + 2y^2 = 1$, we can use the method of Lagrange multipliers or directly solve the constraint for one variable and substitute it into the function. However, in this case, a simpler approach is to notice that both $f(x, y)$ and the constraint involve sums of squares, and we are essentially looking for the points on the ellipse where the sum of squares of $x$ and $y$ is minimized and maximized.Given the constraint $3x^2 + 2y^2 = 1$, we can express $y^2$ in terms of $x^2$ as $y^2 = \frac{1 - 3x^2}{2}$. Substituting this into $f(x, y)$, we get a function of $x$ alone, which can be analyzed for extrema. However, without explicit calculation, we can reason about the problem geometrically and algebraically.The minimum value of $f(x, y)$ occurs when both $x^2$ and $y^2$ are as small as possible given the constraint, which geometrically corresponds to the points on the ellipse closest to the origin. Conversely, the maximum value occurs at the points farthest from the origin, still lying on the ellipse.To find these values explicitly, we can also use the method of substituting one variable and then differentiating, but it's clear from the options provided that we're looking for specific rational fractions, which suggests a more algebraic solution might be intended.Given the constraint, we can see that the minimum and maximum values of $x^2 + y^2$ must correspond to the extremal points of the ellipse where either $x$ or $y$ reaches its extremal value given the constraint. The correct answer involves recognizing that the sum of squares $x^2 + y^2$ is directly related to the distance from the origin in Cartesian coordinates, and the constraint defines an ellipse with a specific relationship between $x^2$ and $y^2$.The minimum and maximum values can be found by considering the ellipse's major and minor axes and how they relate to the function $f(x, y)$. Without solving the system explicitly, the correct choice is based on understanding these geometric relationships and the nature of the problem, which suggests that there are indeed bounded minimum and maximum values for $f(x, y)$ on the ellipse, contrary to option D, and that these values must correspond to specific points on the ellipse where the sum of squares is minimized and maximized, leading to the selection of option B as the correct answer.

Find the Minimum and Maximum Value Of f(x,y)=x2+y2f ( x , y ) = x ^ { 2 } + y ^ { 2 }f(x,y)=x2+y2

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