Practices on Max/Min.

  1. Find the relative extrema for the following functions:

    1. $f(x,y)=x^{3}-3x+y$ [Ans: Set $f_{x}=3x^{2}-3=0$ and $f_{y}=1=0;$ no critical number]

    2. MATH [Ans: no crical number]

    3. MATH [Ans: Set MATH yields, the crical number at $(1,1).$ Since MATH and $f_{xx}(1,1)>0,$ $f$ has a local minimum at $(1,1).$

    4. MATH [Ans: $f$ has a local minimum at $(4,-2)$. ]

    5. MATH [Ans: $f$ has a local minimum at $(1,-2).$ ]

    6. MATH [Ans: $f$ has a relative maximum at MATH.]

    7. MATH [Ans: $f$ has a saddle point at $(0,0)$ and a relative minimum at $(2,2).$ ]

    8. MATH [Ans: $f$ has a saddle point at $(1,\frac{3}{2}),$ and a relative minimum at $(5,\frac{27}{2}).$ ]

    9. MATH [Ans: $f$ has a relative minimum at $(\dfrac{1}{2},-1).$ ]

    10. $f(x,y)=(x+y)(xy+1)$ [Ans: $f$ has saddle points at $(1,-1)$ and $(-1,1).$ ]

  2. Use Maple to sketch the contour map (collections of the level curves for $f(x,y)=k$) for problems (g) through (j) above.

  3. Given the following surface determined by the function $f,$find $\dfrac{df}{dt}$ if $r(t)=(x(t),y(t))$ is a curve on the surface.

    1. $f(x,y)=x-y,$ $r(t)=(at,b\cos at)$. (where $a$ and $b$ are constants). [Ans: MATH ]

    2. $f(x,y,z)=xy-yz,$ MATH [Ans: $3t^{2}-5t^{4}.$ ]

    3. MATH MATH (where $a,b,$and $w$ are constants) [Ans: MATH

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