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

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

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

  2. Let $f(x,y) = 2\,x^2+2\,x^2\,y+y^2$. 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 ] , $\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 ]

  3. Let $\displaystyle \frac{4\,x^2}{61}+\frac{9\,y^2}{61}+\frac{36\,z^2}{61}=1$ be the ellipsoid. Find the tangent plane at the point [ 2 ,   − 1 ,   − 1 ] .

    $\displaystyle \frac{72\,x}{61}-\frac{4\,y}{61}-\frac{9\,z}{61}=1$ $\displaystyle \frac{8\,x}{61}-\frac{9\,y}{61}-\frac{36\,z}{61}=1$ $\displaystyle \frac{18\,x}{61}-\frac{4\,y}{61}-\frac{36\,z}{61}=1$ $\displaystyle \frac{8\,x}{61}-\frac{36\,y}{61}-\frac{9\,z}{61}=1$

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

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

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

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

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

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

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

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

    It is a saddle point It is a local minimum

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

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

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

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

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

  8. 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}

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

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

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

    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

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

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


Department of Mathematics
Last modified: 2026-09-29