Unknown

7.5. Other Problems 229

4.4. Apply the AGM inequality to {rf}.

4.5, 4.6. Take the difference of the two sides.

4.7. If a # 0, let u = p-’ + q-l, v = r-l + s-l. Use the AGM inequality

to obtain a lower estimate for c/u - b/a in terms of u and v.

4.8. Use induction.

4.9. Let c = ua3 and d = vb3. The condition expressed in terms of u, v

and w = u2(u2 + b2)-l is u2w3 + v2(1 - ~1)~ = 1. When ad = bc,

we have that w = v(u + v)-’ and the condition becomes u2v3 +

v2u3 = (u+v)~, which is equivalent to 1 = (u-l +v-~)~. It has to be

shown in general that 21-l +v-’ 1 1. This can be done by looking at

u2w3 + v2(1- w>s - (u-1 + v-l)-?

4.11. Let A, B be acute and express the left side in terms of u = tan A/2
and v = tan B/2; apply the AGM inequality.

4.12. Express the sum of squares in terms of u and minimize over those u
for which the zeros of the quadratic are real.

4.13. Use the AGM inequality for the set which contains x/m m times and
y/n n times.

4.14. The inequality is equivalent to y 5 min{x + 2/x : z 2 0). The
function in the brackets can be minimized using the AGM inequality
in a way similar to Problem 13.

4.15. Let f(t) = (t - r)g(t); note that g(r) # 0 (why?). Determine M so
that Is(t)] 5 M for It - r] 5 1 and observe that If(p/q)I > l/q”.

5.1. Consider the coefficient of P-l of the Lagrange polynomial p(t) for

which p(ai) = f(oi). What is p(t)?

5.2. Use Lagrange polynomials.

5.3. Use double induction on k and n. The cases n = 1 and k = 1 are

clear. Suppose the result is known for any n and 1 5 k 5 r - 1 as

wellasfork=rand l<n<m-1.

5.4. f(m) is the nth order difference of a polynomial of degree n. Write

the sum without the summation sign.

5.5. If l/2 2 ]u] 5 1, write g(x) as a Lagrange polynomial with respect

to its values at x = -1, 0, 1. Then one can get a sharp estimate of

]g’(u)] in terms of ]g(-l)], ]g(O)] and ]g(l)]. The case 0 < ]u] 5 l/2

is more difficult. Sketch a few graphs; there are essentially two cases

to consider according as g(z) is monotone on [-1, l] or not. Reduce

to the situation that g(z) assumes both the values -1 and +l.