I've been having some trouble with this one:
Prove that if $\displaystyle x^2$ > k(x - 2) for all real x, then 0 < k < 8.
I started with taking the k(x-2) to the other side, but I don't think I'm going anywhere with this.
Thanks in advance.
I've been having some trouble with this one:
Prove that if $\displaystyle x^2$ > k(x - 2) for all real x, then 0 < k < 8.
I started with taking the k(x-2) to the other side, but I don't think I'm going anywhere with this.
Thanks in advance.
$\displaystyle \displaystyle \begin{align*} x^2 &> k(x - 2) \\ x^2 &> kx - 2k \\ x^2 - kx + 2k &> 0\end{align*}$
For this inequation to be true, then the discriminant of $\displaystyle \displaystyle x^2 - kx + 2k$ must be negative.
So
$\displaystyle \displaystyle \begin{align*} (-k)^2 - 4(1)(2k) &< 0 \\ k^2 - 8k &< 0 \\ k^2 - 8k + (-4)^2 &< (-4)^2 \\ (k - 4)^2 &< 16 \\ |k - 4| &< 4 \\ -4 < k - 4 &< 4 \\ 0 < k &< 8 \end{align*}$
$\displaystyle x^2-kx+2k>0$
is true for all x, if the U-shaped quadratic never touches the x-axis,
so the equation
$\displaystyle x^2-kx+2k=0$
is of the form
$\displaystyle ax^2+bx+c=0$
and must have complex roots
$\displaystyle \Rightarrow\ b^2-4ac<0\Rightarrow\ k^2-8k<0$
$\displaystyle \Rightarrow\ k(k-8)<0$
One factor is positive and the other is negative.
k>8 and k<0 cannot occur, so k>0 and k<8.