1.2. Quadratic Polynomials 11
(b) Find a counterexample to (a) if the word “nonrational” is re-
placed by “rational.”
- 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.
- (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?
- (a) Verify the Lagrange identity:
(b) Use (a) to establish the Cauchy-Schwarz-Bunjakovsky Inequal-
ity.
- 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.