Advanced High-School Mathematics

152 CHAPTER 3 Inequalities

6. Find the maximum value of the objective functionx+y+zgiven
   thatx^2 +y^2 +z^2 = 4. (Hint: use AM(x,y,z)≤QM(x,y,z).) Can
   you describe this situation geometrically?


7. Find the maximum value of the objective function x^2 +y^2 +z^2
   given thatx+y+z= 6.


8. Suppose thatxandyare positive numbers with
   x+y= 1. Show that

1

x

+

1

y

≥4.

9. Suppose thatxandyare positive numbers with
   x+y= 1. Compute the minimum value of

(

1 +

1

x

) (

1 +

1

y

)

. (This
was already given as Exercise 4 on page 147. However, doesn’t it
really belong in this section? Can you relate it to Exercise 8,
above?)
10. Assume thatx 1 , x 2 , ..., xn>0 and thatx 1 +···+xn= 1. Prove
that
1
x 1

+···+

1

xn

≥n^2.

(Hint: don’t use mathematical induction!)

11. Letn≥ 2 , x, y >0. Show that

2

n∑− 1

k=1

xkyn−k≤(n−1)(xn+yn).

(This is somewhat involved; try arguing along the following lines.

(i) Let P(x,y) = (n −1)(xn +yn)− 2

n∑− 1

k=1

xkyn−k; note that

P(y,y) = 0 (i.e., x = y is a zero of P(x,y) regarded as a

polynomial inx).

(ii) Show that

d

dx

P(x,y)

∣∣

∣∣

∣x=y = 0. Why does this show that

P(x,y) has at least a double zero atx=y?

(iii) Use Descartes Rule of Signs to argue that P(x,y) has, for

x, y > 0 onlya double zero atx=y.