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Bounded homework

  1. Jan 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Let [tex]f:[-1,1] \times \mathbb{R} \to\mathbb{R}[/tex] be a function.

    If [itex]f[/itex] is defined by:

    (i) [itex]f(x,y) = 3\exp(x-y^2)[/itex]

    then is the derivative with respect to [itex]y[/itex] bounded?

    If [itex]f[/itex] is defined by:

    (ii) [itex]f(x,y) = 7\exp(y^2-x)[/itex]

    then is the derivative with respect to [itex]y[/itex] bounded?

    3. The attempt at a solution

    For (i):

    [itex]\frac{\partial f}{\partial y} = -6y\exp(x-y^2) = -6y\exp(-y^2)\exp(x)[/itex]

    and since [itex]\exp(x) \leq e[/itex] in the specified domain and since [itex]y\exp(-y^2)[/itex] is a bounded function on [itex]\mathbb{R}[/itex], [itex]\frac{\partial f}{\partial y}[/itex] is bounded - right?

    For (ii):

    [itex]\frac{\partial f}{\partial y} = 14y\exp(y^2-x) = 14y\exp(y^2)\exp(-x)[/itex]

    This is not bounded - right?
    Last edited: Jan 25, 2012
  2. jcsd
  3. Jan 25, 2012 #2


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    Re: Bounded?

    That looks right.
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