1. 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[ 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] $

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

  2. 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 $\displaystyle \frac{13}{4}$ 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 $\displaystyle \frac{17}{4}$ at $\displaystyle \left[ y=\frac{7}{4} , z=\frac{1}{2} , x=1 \right] $

  3. Suppose that $\displaystyle \frac{x}{19}+\frac{9\,y}{38}-\frac{9\,z}{19}=1$ is the tangent plane at the point [ 1 ,  2 ,   − 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 $c$.

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

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

    [ 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 ] [ 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] $

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

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

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

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

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

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

    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}

  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.

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

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

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

    The extreme value $\displaystyle \frac{3}{31}$ at $\displaystyle \left[ z=\frac{9}{31} , y=\frac{3}{31} , x=\frac{1}{31} \right] $

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

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

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


Department of Mathematics
Last modified: 2026-08-09