Unknown

1.2. Quadratic Polynomials 11

(b) Find a counterexample to (a) if the word “nonrational” is re-

placed by “rational.”

13. Show that, if 2 = (b - d)/(a - c) satisfies one of the equations
    x2 - ux+b=O

x2-cx+d=O,

then it satisfies the other as well.

14. (a) Let ei, bi be nonnegative reals (1 5 i 5 n). The function

e(uit + bi)2

i=l

is a polynomial in t. Explain why its discriminant is nonpositive.

(b) Use (a) to establish the Cauchy-Schwarz-Bunjakovsky Inequal-

ity:

When does equality occur?

15. (a) Verify the Lagrange identity:

(b) Use (a) to establish the Cauchy-Schwarz-Bunjakovsky Inequal-

ity.

16. Diameters of an ellipse. The equation of an ellipse whose axes lie
    along the axes of coordinates and whose center is at the origin can
    be written b2x2 + .2y2 = b2a2, where a and b are the lengths of the
    semi-axes. Find the locus of the midpoints of chords of the ellipse
    with fixed slope m; such a locus is called a diameter of the ellipse.