Generating...                               quiz05c_n20

  1. Find the extreme value for $f(x,y) = e^{3\,x\,y}$ subject to $\displaystyle x^3+y^3=54$ .

    The extreme value $\displaystyle e^9$ at x = 3 and y = 3

    The extreme value $\displaystyle e^4$ at x = 2 and y = 2

    The extreme value $\displaystyle e^{12}$ at x = 2 and y = 2

    The extreme value $\displaystyle e^{27}$ at x = 3 and y = 3

  2. Find the extreme value for $f(x,y,z) = 2\,x+y+z$ subject to $\displaystyle x^2+y+z^2=3$ .

    The extreme value $\displaystyle \frac{11}{2}$ at $\displaystyle \left[ y=\frac{1}{2} , z=\frac{1}{2} , x=\frac{3}{2} \right] $

    The extreme value $\displaystyle \frac{9}{2}$ at $\displaystyle \left[ y=-\frac{1}{2} , z=\frac{1}{2} , x=\frac{3}{2} \right] $

    The extreme value $\displaystyle \frac{17}{4}$ at $\displaystyle \left[ y=\frac{7}{4} , z=\frac{1}{2} , x=1 \right] $

    The extreme value $\displaystyle \frac{13}{4}$ at $\displaystyle \left[ y=\frac{3}{4} , z=\frac{1}{2} , x=1 \right] $

  3. Let $f(x,y) = x^2+x^3+3\,y^2+x\,y^2$. Find the value $D$ at the critical point [ y = 0 ,  x = 0 ] for second derivative test.

    $D = 12$ $D = -\frac{28}{3}$ $D = 16$ $D = -\frac{16}{3}$

  4. Choose the correct statement regarding the extreme value for $f(x,y) = e^{3\,x\,y}$ subject to $\displaystyle x^3+y^3=2$ .

    The point $(x,y)$ at the extreme value satisfies \begin{displaymath}\begin{cases} 3\,x\,y\,e^{3\,x\,y} = \lambda ( 3\,y^2); \\ ... ...^{3\,x\,y} = \lambda ( 3\,x^2); \\ x^3+y^3 = 2. \end{cases}\end{displaymath}

    The point $(x,y)$ at the extreme value satisfies \begin{displaymath}\begin{cases} e^{3\,x\,y} = \lambda ( 3\,x^2); \\ e^{3\,x\,y} = \lambda ( 3\,y^2); \\ x^3+y^3 = 2. \end{cases}\end{displaymath}

    The point $(x,y)$ at the extreme value satisfies \begin{displaymath}\begin{cases} 3\,y\,e^{3\,x\,y} = \lambda ( x^3+y^3); \\ 3... ...{3\,x\,y} = \lambda ( x^3+y^3); \\ x^3+y^3 = 2. \end{cases}\end{displaymath}

    The point $(x,y)$ at the extreme value satisfies \begin{displaymath}\begin{cases} 3\,y\,e^{3\,x\,y} = \lambda ( 3\,x^2); \\ 3\... ...^{3\,x\,y} = \lambda ( 3\,y^2); \\ x^3+y^3 = 2. \end{cases}\end{displaymath}

  5. Suppose that $\displaystyle \frac{36\,x}{49}+\frac{9\,y}{49}+\frac{4\,z}{49}=1$ is the tangent plane at the point [ 1 ,  1 ,  1 ] to the ellipsoid $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1$ . Then find the values $a$, $b$ and $c$.

    $a = \frac{7}{6}$, $b = \frac{7}{3}$, and $c = \frac{7}{2}$. $a = \frac{7}{2}$, $b = \frac{7}{3}$, and $c = \frac{7}{6}$. $a = \frac{7}{3}$, $b = \frac{7}{6}$, and $c = \frac{7}{2}$. $a = \frac{7}{6}$, $b = \frac{7}{2}$, and $c = \frac{7}{3}$.

  6. Find the extreme value for $f(x,y,z) = 3\,x^2+y^2+z^2$ subject to x + y + z = 3 .

    The extreme value $\displaystyle \frac{18}{5}$ at $\displaystyle \left[ z=\frac{6}{5} , y=\frac{6}{5} , x=\frac{3}{5} \right] $

    The extreme value $\displaystyle \frac{27}{7}$ at $\displaystyle \left[ z=\frac{9}{7} , y=\frac{9}{7} , x=\frac{3}{7} \right] $

    The extreme value $\displaystyle \frac{8}{5}$ at $\displaystyle \left[ z=\frac{4}{5} , y=\frac{4}{5} , x=\frac{2}{5} \right] $

    The extreme value $\displaystyle \frac{12}{7}$ at $\displaystyle \left[ z=\frac{6}{7} , y=\frac{6}{7} , x=\frac{2}{7} \right] $

  7. Let $f(x,y) = x^2+x^3+2\,y^2+x\,y^2$. Find the gradient $\nabla f(x,y)$ .

    $\displaystyle \left[ 2\,x+3\,x^2+y^2 , 6\,y+2\,x\,y \right] $

    $\displaystyle \left[ 2\,x+3\,x^2+y^2 , 4\,y+2\,x\,y \right] $

    $\displaystyle \left[ 4\,x+3\,x^2+y^2 , 4\,y+2\,x\,y \right] $

    $\displaystyle \left[ 4\,x+3\,x^2+y^2 , 6\,y+2\,x\,y \right] $

  8. Let $f(x,y) = x^2+2\,x^2\,y+y^2$. Find the gradient $\nabla f(x,y)$ .

    $\displaystyle \left[ 4\,x+2\,x\,y , x^2+2\,y \right] $

    $\displaystyle \left[ 4\,x+4\,x\,y , 2\,x^2+2\,y \right] $

    $\displaystyle \left[ 2\,x+4\,x\,y , 2\,x^2+2\,y \right] $

    $\displaystyle \left[ 2\,x+2\,x\,y , x^2+2\,y \right] $

  9. Let $\displaystyle \frac{4\,x^2}{61}+\frac{9\,y^2}{61}+\frac{36\,z^2}{61}=1$ be the ellipsoid. Find the tangent plane at the point [ 2 ,   − 1 ,  1 ] .

    $\displaystyle \frac{8\,x}{61}-\frac{9\,y}{61}+\frac{36\,z}{61}=1$ $\displaystyle \frac{18\,x}{61}-\frac{36\,y}{61}+\frac{4\,z}{61}=1$ $\displaystyle \frac{72\,x}{61}-\frac{4\,y}{61}+\frac{9\,z}{61}=1$ $\displaystyle \frac{18\,x}{61}-\frac{4\,y}{61}+\frac{36\,z}{61}=1$

  10. Choose the correct statement regarding the critical points for $f(x,y) = 2\,x^2+x^3+3\,y^2+x\,y^2$.

    The critical points $(x,y)$ satisfy \begin{displaymath}\begin{cases} 2\,x+3\,x^2+y^2 = 0; \\ 6\,y+2\,x\,y = 0. \end{cases}\end{displaymath}

    The critical points $(x,y)$ satisfy \begin{displaymath}\begin{cases} 4\,x+3\,x^2+y^2 = 0; \\ 4\,y+2\,x\,y = 0. \end{cases}\end{displaymath}

    The critical points $(x,y)$ satisfy \begin{displaymath}\begin{cases} 2\,x+3\,x^2+y^2 = 0; \\ 4\,y+2\,x\,y = 0. \end{cases}\end{displaymath}

    The critical points $(x,y)$ satisfy \begin{displaymath}\begin{cases} 4\,x+3\,x^2+y^2 = 0; \\ 6\,y+2\,x\,y = 0. \end{cases}\end{displaymath}


Department of Mathematics
Last modified: 2026-05-20