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Math Help - function problem

  1. #1
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    function problem

    I am having problem doing this problem it is asking to find the condition that must be satisfied by k in order that the expression 2x^2 +6x +1 +k(x^2+2) may be positive for all real values of x.
    Can some help me with this problem?
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  2. #2
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    Quote Originally Posted by scrible View Post
    I am having problem doing this problem it is asking to find the condition that must be satisfied by k in order that the expression 2x^2 +6x +1 +k(x^2+2) may be positive for all real values of x.
    Can some help me with this problem?

    That's a parabola , so it'll be always positive iff (1) it is a "smiling" or convex upwards parabola (i.e., its higher coefficient is positive), and (2) its discriminant is negative, i.e.: iff it has no real roots, so:

    2x^2 +6x +1 +k(x^2+2)\Longrightarrow (k+2)x^2+6x+(2k+1)>0\,\,\forall\,x\in \mathbb{R}\Longleftrightarrow \,(1)\,\,k+2>0\,\,and\,\,(2)\,\,\Delta=b^2-4ac=6^2-4(k+2)(2k+1)<0:

    (1)\,\,k+2>0\,\Longrightarrow\,k>-2\,;

    (2)\,\,\Delta=36-8k^2-20k-8<0\,\Longrightarrow \,-8k^2-20k+28<0\,\,(divide\,\,by\,\,-4)\,\,\Longrightarrow\,2k^2+5k-7>0 \Longrightarrow\,(2k+7)(k-1)>0\,\Longrightarrow\,k<-\frac{7}{2}\,\,or\,\,k>1

    Taking together both conditions for k on (1)-(2) the solution is immediate.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    That's a parabola , so it'll be always positive iff (1) it is a "smiling" or convex upwards parabola (i.e., its higher coefficient is positive), and (2) its discriminant is negative, i.e.: iff it has no real roots, so:

    2x^2 +6x +1 +k(x^2+2)\Longrightarrow (k+2)x^2+6x+(2k+1)>0\,\,\forall\,x\in \mathbb{R}\Longleftrightarrow \,(1)\,\,k+2>0\,\,and\,\,(2)\,\,\Delta=b^2-4ac=6^2-4(k+2)(2k+1)<0:

    (1)\,\,k+2>0\,\Longrightarrow\,k>-2\,;

    (2)\,\,\Delta=36-8k^2-20k-8<0\,\Longrightarrow \,-8k^2-20k+28<0\,\,(divide\,\,by\,\,-4)\,\,\Longrightarrow\,2k^2+5k-7>0 \Longrightarrow\,(2k+7)(k-1)>0\,\Longrightarrow\,k<-\frac{7}{2}\,\,or\,\,k>1

    Taking together both conditions for k on (1)-(2) the solution is immediate.

    Tonio
    Thank you very much
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