This is a nice problem. Okay, so we have f: y = x^4 - 2; and g: y = kx^2. We have to show that they intersect for all values of k. Intersect means that cross at least once, and so we are going to have to show that for each value of k, f and g intersect at least once.
Let's start by seeing where there is an intersection between f and g, if they do intersect. We do this by setting them equal:
x^4 - 2 = kx^2; and simplifying gives us h: x^4 - kx^2 - 2 = 0, giving us a quadratic equation in x^2. Now we could use the quadratic formula here to actually try to generate a solution, in which you'd find that
x^2 = (k +/- sqrt(k^2 + 8))/2;
but Descartes rule of signs is probably a little more straight forward in this case, since we do not need to actually show what the solution is to h(x). Here we have
h(x) = h(-x) = x^4 - kx^2 - 2,
which has 1 sign change in the positive case and 1 sign change in the negative. Thus h will always have 1 positive real root, and 1 negative real root, and this is independent of the value of k. Therefore, we know that there are always real roots for h, meaning that f and g always intersect, regardless of the value of k.
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