Generating...                               quiz05c_n11

  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.

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

  2. Choose the correct answer regarding the critical point [ y = 0 ,  x = 0 ] for $f(x,y) = 2\,x^2+x^3+2\,y^2+x\,y^2$.

    It is a local minimum

    It is a saddle point

  3. Let $\displaystyle \frac{x^2}{19}+\frac{9\,y^2}{76}+\frac{9\,z^2}{19}=1$ be the ellipsoid. Find the tangent plane at the point [ 1 ,  2 ,   − 1 ] .

    $\displaystyle \frac{x}{19}+\frac{9\,y}{38}-\frac{9\,z}{19}=1$ $\displaystyle \frac{9\,x}{76}+\frac{2\,y}{19}-\frac{9\,z}{19}=1$ $\displaystyle \frac{9\,x}{76}+\frac{18\,y}{19}-\frac{z}{19}=1$ $\displaystyle \frac{9\,x}{19}+\frac{9\,y}{38}-\frac{z}{19}=1$

  4. 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.

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

  5. Choose the correct answer regarding the critical point [ y = 0 ,  x = 0 ] for $f(x,y) = 2\,x^2+x^3+2\,y^2+x\,y^2$.

    Since $D = -\frac{16}{3}$ and $f_{xx} = -4$, it is a saddle point

    Since $D = -\frac{16}{3}$ and $f_{xx} = -4$, it is a local maximum

    Since $D = 16$ and $f_{xx} = 4$, it is a local minimum

    Since $D = 16$ and $f_{xx} = 4$, it is a saddle point

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

    The extreme value $\displaystyle \frac{18}{11}$ at $\displaystyle \left[ z=\frac{12}{11} , y=\frac{6}{11} , x=\frac{3}{11} \right] $

    The extreme value $\displaystyle \frac{8}{11}$ at $\displaystyle \left[ z=\frac{8}{11} , y=\frac{4}{11} , x=\frac{2}{11} \right] $

    The extreme value $\displaystyle \frac{3}{4}$ at $\displaystyle \left[ z=\frac{3}{4} , y=\frac{3}{8} , x=\frac{1}{8} \right] $

    The extreme value $\displaystyle \frac{27}{16}$ at $\displaystyle \left[ z=\frac{9}{8} , y=\frac{9}{16} , x=\frac{3}{16} \right] $

  7. Choose the correct statement regarding the extreme value for $f(x,y) = e^{x\,y}$ subject to $\displaystyle x^3+y^3=54$ .

    The point $(x,y)$ at the extreme value satisfies \begin{displaymath}\begin{cases} y\,e^{x\,y} = \lambda ( x^3+y^3); \\ x\,e^{x\,y} = \lambda ( x^3+y^3); \\ x^3+y^3 = 54. \end{cases}\end{displaymath}

    The point $(x,y)$ at the extreme value satisfies \begin{displaymath}\begin{cases} e^{x\,y} = \lambda ( 3\,x^2); \\ e^{x\,y} = \lambda ( 3\,y^2); \\ x^3+y^3 = 54. \end{cases}\end{displaymath}

    The point $(x,y)$ at the extreme value satisfies \begin{displaymath}\begin{cases} x\,y\,e^{x\,y} = \lambda ( 3\,y^2); \\ x\,y\,e^{x\,y} = \lambda ( 3\,x^2); \\ x^3+y^3 = 54. \end{cases}\end{displaymath}

    The point $(x,y)$ at the extreme value satisfies \begin{displaymath}\begin{cases} y\,e^{x\,y} = \lambda ( 3\,x^2); \\ x\,e^{x\,y} = \lambda ( 3\,y^2); \\ x^3+y^3 = 54. \end{cases}\end{displaymath}

  8. Let $f(x,y) = 2\,x^2+x^2\,y+y^2$. Find all the critical points.

    [ y = 0 ,  x = 0 ] , $\displaystyle \left[ y=-\frac{1}{2} , x=-\frac{1}{\sqrt{2}} \right] $ and $\displaystyle \left[ y=-\frac{1}{2} , x=\frac{1}{\sqrt{2}} \right] $ [ y = 0 ,  x = 0 ] , $\displaystyle \left[ y=-1 , x=-\sqrt{2} \right] $ and $\displaystyle \left[ y=-1 , x=\sqrt{2} \right] $ [ y = 0 ,  x = 0 ] , [ y = −2 ,  x = 2 ] and [ y = −2 ,  x =  − 2 ] [ y = 0 ,  x = 0 ] , [ y = −1 ,  x =  − 1 ] and [ y = −1 ,  x = 1 ]

  9. 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; \\ 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}

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

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

    The extreme value $\displaystyle \frac{5}{2}$ at $\displaystyle \left[ y=\frac{3}{2} , z=\frac{1}{2} , x=\frac{1}{2} \right] $


Department of Mathematics
Last modified: 2026-02-06