Advanced High-School Mathematics

172 CHAPTER 3 Inequalities

Example. Compute the minimum value of the function

f(x) =

1

x^2

+x, x >0.

Solution. The minimum value will occur where the line y = c is

tangent to the graph of y = f(x). We may write the equation

f(x) =cin the form of a cubic polynomial inx:

x^3 −cx^2 + 1 = 0.

As the tangent indicates a multiple zero, we must have ∆ = 0. As

a= 1, b=−c, c= 0, d= 1, we get the equation 4c^3 −27 = 0,

which implies that the minimum value is given byc=

3

√ 34 (which

can be verified by standard calculus techniques).

Now try these:

Exercises.

1. Compute the minimum of the function
   h(x) =

1

x

+x^2 , x >0.

2. Compute the minimum of 2y−xgiven that
   x^3 −x^2 y+ 1 = 0.


3. Compute the maximum value ofy+x^2 given that
   x^3 −xy+ 2 = 0.