Generating...                               quiz05c_n4

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

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

    $D = 24$ $D = 8$ $D = -\frac{40}{3}$ $D = -\frac{16}{3}$

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

    $\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] $

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

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

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

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

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

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

    The extreme value $\displaystyle e^3$ at x = 1 and y = 1

  6. 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 ( 3\,x^2); \\ 3\... ...^{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}

  7. Let $\displaystyle \frac{9\,x^2}{184}+\frac{y^2}{46}+\frac{9\,z^2}{46}=1$ be the ellipsoid. Find the tangent plane at the point [ 2 ,  1 ,  2 ] .

    $\displaystyle \frac{9\,x}{92}+\frac{y}{46}+\frac{9\,z}{23}=1$ $\displaystyle \frac{x}{23}+\frac{9\,y}{184}+\frac{9\,z}{23}=1$ $\displaystyle \frac{9\,x}{23}+\frac{y}{46}+\frac{9\,z}{92}=1$ $\displaystyle \frac{9\,x}{92}+\frac{9\,y}{46}+\frac{z}{23}=1$

  8. Suppose that $\displaystyle -\frac{9\,x}{19}+\frac{y}{19}+\frac{9\,z}{38}=1$ 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 $c$.

    $a = \sqrt{19}$, $b = \frac{\sqrt{19}}{3}$, and $c = \frac{2\,\sqrt{19}}{3}$. $a = \frac{2\,\sqrt{19}}{3}$, $b = \frac{\sqrt{19}}{3}$, and $c = \sqrt{19}$. $a = \frac{2\,\sqrt{19}}{3}$, $b = \sqrt{19}$, and $c = \frac{\sqrt{19}}{3}$. $a = \frac{\sqrt{19}}{3}$, $b = \sqrt{19}$, and $c = \frac{2\,\sqrt{19}}{3}$.

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

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

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

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

  10. 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 $\displaystyle \frac{11}{2}$ 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 $\displaystyle \frac{5}{2}$ at $\displaystyle \left[ y=\frac{3}{2} , z=\frac{1}{2} , x=\frac{1}{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] $


Department of Mathematics
Last modified: 2025-10-30