Question

1.What is the smallest possible value of 2x^4+(8/x^4) over all real nonzero values of x? To find the minimum value you have to factor it, but I am not sure how to factor this.

2.What is the maximum value of c such that the graph of the parabola y=1/3x^2 has at most one point of intersection with the line y=x+c? I am not sure how to start this, how do we find the maximum value and what is c?

Thank you!

Answer 1

2x^4 + 8/x^4
= 2(x^4 + 4/x^4)
= 2(x^4 - 4 + 4/x^4) + 8
= 2(x^2 - 2/x^2)^2 + 8
This has its minimum value of 8 when
x^2 - 2/x^2 = 0
x^4 - 2 = 0
x = ±∜2
-----------------------------
If there is only one point of intersection with the line y=x+c, then consider the function

y = 1/3 x^2 - (x+c)
That will have only one root, which means the discriminant is zero. So,

y = 1/3 x^2 - x - c
the discriminant is 1+4c/3, which is zero when c = -3/4

See the graphs at

http://www.wolframalpha.com/input/?i=plot+y+%3D+1%2F3+x^2%2C+y+%3D+x-3%2F4+for+x%3D0..3