Advanced High-School Mathematics

166 CHAPTER 3 Inequalities

Therefore, the maximum value of x+y is 3 (and the minimum

value ofx+yis−3).

Exercises.

1. Given thatx+y= 1, x, y >0, find the minimum value of

1

x

+

1

y

.

2. Given that

1

x

+

1

y

= 1, x, y >0, prove thatx+y≥4.

(Exercises 1 and 2 can be solved very simply by multiplying to-

gether

1

x

+

1

y

andx+yand using the result of Example 2.)

3. Find the distance from the origin to the line with equation
   x+ 3y= 6.


4. Given that

x

y

+y= 1 find the minimum value of

x+y, x, y >0.

5. Find the largest value of a so that the parabola with equation
   y=a−x^2 is tangent to the circle with graphx^2 +y^2 = 4. Go on
   to argue that this value ofais the maximum of the functionx^2 +y
   given thatx^2 +y^2 = 4.


6. Letf(x) =ax^2 +bx+c, and so the derivative is f′(x) = 2ax+b.
   Denote byR(f) the determinant

R(f) = det







a b c

2 a b 0

0 2a b





.

Show thatR(f) =−aD(f),whereD=D(f) is the discriminant

off(x).