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Thread: Inequalities

  1. #1
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    Inequalities

    The equation of a curve is $\displaystyle x^2y^2-x^2+y^2=0$
    a) find the equation of the tangents at the origin
    b) find the equations of the real asymptotes
    c) show that the numerical value of y is never greater than the corresponding value of x
    d)show that the numerical value of y is always less than unity

    My problem is with (c) and (d). I've done (a) and (b) already.
    For (c), $\displaystyle x-y\geq 0$
    $\displaystyle y=\pm\frac{x}{\sqrt{x^2+1}}$
    Substitute
    $\displaystyle x-\frac{x}{\sqrt{x^2+1}}=\frac{x\sqrt{x^2+1}-x}{\sqrt{x^2+1}}$
    Now I don't know how to continue to make it such that it is greater or equal to 0, which then i can show that y is always lesser of equal to x.
    Same problem with the next one. The $\displaystyle \sqrt{x^2+1}$ is what stumble me.
    Thanks!
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  2. #2
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    I think you should rewrite this as

    $\displaystyle y = \left|\frac{x}{\sqrt{x^2 + 1}}\right|$.


    Then to show that $\displaystyle y \leq x$, you would have

    $\displaystyle \left|\frac{x}{\sqrt{x^2 + 1}}\right| \leq x$

    $\displaystyle -x \leq \frac{x}{\sqrt{x^2 + 1}} \leq x$

    $\displaystyle -1 \leq \frac{1}{\sqrt{x^2 + 1}}\leq 1$.


    This would only ever not be true if the denominator was smaller than the numerator. So it would not be true if $\displaystyle \sqrt{x^2 + 1} < 1$

    If $\displaystyle x^2 + 1 < 1$ then

    $\displaystyle x^2 < 0$.

    Since this is never true, that means that

    $\displaystyle -1 \leq \frac{1}{\sqrt{x^2 + 1}} \leq 1$ IS true.

    Therefore $\displaystyle y \leq x$ for all $\displaystyle x$.


    To answer d) you want to show that

    $\displaystyle y = \left|\frac{x}{\sqrt{x^2 + 1}}\right| < 1$.


    Therefore $\displaystyle -1 < \frac{x}{\sqrt{x^2 + 1}} < 1$

    $\displaystyle -\sqrt{x^2 + 1} < x < \sqrt{x^2 + 1}$


    This should be obvious, since $\displaystyle x = \sqrt{x^2}$.

    $\displaystyle \sqrt{x^2} < \sqrt{x^2 + 1}$.
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