Generating...                               quiz05c_n28

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

    $D = 8$ $D = -16$ $D = 4$ $D = -8$

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

    The extreme value $\displaystyle \frac{9}{11}$ at $\displaystyle \left[ z=\frac{9}{11} , y=\frac{3}{11} , x=\frac{3}{11} \right] $

    The extreme value $\displaystyle \frac{12}{31}$ at $\displaystyle \left[ z=\frac{18}{31} , y=\frac{6}{31} , x=\frac{2}{31} \right] $

    The extreme value $\displaystyle \frac{4}{11}$ at $\displaystyle \left[ z=\frac{6}{11} , y=\frac{2}{11} , x=\frac{2}{11} \right] $

    The extreme value $\displaystyle \frac{27}{31}$ at $\displaystyle \left[ z=\frac{27}{31} , y=\frac{9}{31} , x=\frac{3}{31} \right] $

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

    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}

    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; \\ 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; \\ 6\,y+2\,x\,y = 0. \end{cases}\end{displaymath}

  4. Let $\displaystyle \frac{36\,x^2}{157}+\frac{4\,y^2}{157}+\frac{9\,z^2}{157}=1$ be the ellipsoid. Find the tangent plane at the point [ 2 ,   − 1 ,   − 1 ] .

    $\displaystyle \frac{18\,x}{157}-\frac{36\,y}{157}-\frac{4\,z}{157}=1$ $\displaystyle \frac{18\,x}{157}-\frac{4\,y}{157}-\frac{36\,z}{157}=1$ $\displaystyle \frac{72\,x}{157}-\frac{4\,y}{157}-\frac{9\,z}{157}=1$ $\displaystyle \frac{72\,x}{157}-\frac{9\,y}{157}-\frac{4\,z}{157}=1$

  5. Choose the correct answer regarding the critical point $\displaystyle \left[ y=0 , x=-\frac{2}{3} \right] $ for $f(x,y) = x^2+x^3+3\,y^2+x\,y^2$.

    It is a saddle point

    It is a local maximum

  6. Choose the correct statement regarding the extreme value for $f(x,y) = e^{3\,x\,y}$ subject to $\displaystyle x^3+y^3=16$ .

    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 = 16. \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 = 16. \end{cases}\end{displaymath}

    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 = 16. \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 = 16. \end{cases}\end{displaymath}

  7. Find the extreme value for $f(x,y,z) = 3\,x+y+z$ subject to $\displaystyle x^2+y+z^2=1$ .

    The extreme value $\displaystyle \frac{13}{4}$ at $\displaystyle \left[ y=\frac{3}{4} , z=\frac{1}{2} , x=1 \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 $\displaystyle \frac{7}{2}$ at $\displaystyle \left[ y=-\frac{3}{2} , z=\frac{1}{2} , x=\frac{3}{2} \right] $

    The extreme value $\displaystyle \frac{9}{4}$ at $\displaystyle \left[ y=-\frac{1}{4} , z=\frac{1}{2} , x=1 \right] $

  8. Let $f(x,y) = x^2+x^3+3\,y^2+x\,y^2$. Find the value $D$ at the critical point $\displaystyle \left[ y=0 , x=-\frac{2}{3} \right] $ for second derivative test.

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

  9. 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[ 2\,x+4\,x\,y , 2\,x^2+2\,y \right] $

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

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

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

    $\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] $

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


Department of Mathematics
Last modified: 2025-10-30