Generating...                               quiz05c_n2

  1. Suppose that $\displaystyle \frac{9\,x}{61}+\frac{36\,y}{61}+\frac{8\,z}{61}=1$ is the tangent plane at the point [ 1 ,  1 ,  2 ] 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 $c$.

    $a = \frac{\sqrt{61}}{3}$, $b = \frac{\sqrt{61}}{2}$, and $c = \frac{\sqrt{61}}{6}$. $a = \frac{\sqrt{61}}{2}$, $b = \frac{\sqrt{61}}{3}$, and $c = \frac{\sqrt{61}}{6}$. $a = \frac{\sqrt{61}}{3}$, $b = \frac{\sqrt{61}}{6}$, and $c = \frac{\sqrt{61}}{2}$. $a = \frac{\sqrt{61}}{2}$, $b = \frac{\sqrt{61}}{6}$, and $c = \frac{\sqrt{61}}{3}$.

  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 $\displaystyle \frac{27}{31}$ at $\displaystyle \left[ z=\frac{27}{31} , y=\frac{9}{31} , x=\frac{3}{31} \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 $\displaystyle \frac{9}{11}$ at $\displaystyle \left[ z=\frac{9}{11} , y=\frac{3}{11} , x=\frac{3}{11} \right] $

  3. Let $f(x,y) = x^2+x^2\,y+y^2$. 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 ] , [ y = −1 ,  x =  − 1 ] and [ y = −1 ,  x = 1 ] [ 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 ]

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

    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}

  5. Find the extreme value for $f(x,y) = e^{x\,y}$ subject to $\displaystyle x^3+y^3=16$ .

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

    The extreme value $\displaystyle e^2$ at x = 1 and y = 1

    The extreme value e at x = 1 and y = 1

    The extreme value $\displaystyle e^8$ at x = 2 and y = 2

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

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

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

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

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

    Since $D = 12$ and $f_{xx} = 2$, it is a saddle point

    Since $D = 12$ and $f_{xx} = 2$, it is a local minimum

    Since $D = -\frac{28}{3}$ and $f_{xx} = -2$, it is a saddle point

    Since $D = -\frac{28}{3}$ and $f_{xx} = -2$, it is a local maximum

  9. Find the extreme value for $f(x,y,z) = 3\,x+y+z$ 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{5}{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] $

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

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

    $\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] $

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

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


Department of Mathematics
Last modified: 2026-07-16