Advanced High-School Mathematics

184 CHAPTER 3 Inequalities

y=3/x+x

y=1/x

3

y=x

3

y=m

We let m be the minimum value of f(x) and note that the graph of

y=mmust be tangent to the graph ofy=

3

x

+x^3. This is equivalent

to saying that the solution of 3 +x^4 =mxis a multiple root, forcing the
discriminant of the quartic polynomialq(x) =x^4 −mx+ 3 to be zero.
That is to say, we need to find that value ofmmaking the discriminant
equal to 0. From the above, we have that

0 = ∆(q) = 4^433 − 33 m^4 = 0⇒m= 4.

In other words, the minimum value ofq(x) on the interval (0,∞) is 4.
(The reader should check that the same result is obtained via differential
calculus.)