Question

Let f1, f2, f3: [a,b] -->R be nonnegative concave functions such that f1(a) = f2(a) = f3(a) = f1(b) = f2(b) = f3(b) = 0. Suppose that max(f1) <= max(f2) <= max(f3).

Prove that: max(f1) + max(f2) <= max(f1+f2+f3)

Answer #1

Prove the following identities. (a) F1 +F3 +F5 +...+F2n−1 = F2n.
(b) F0 −F1 +F2 −F3 +...−F2n−1 +F2n = F2n−1 −1. (c) F02 +F12 +F2
+...+Fn2 = Fn ·Fn+1. (d) Fn−1Fn+1 − Fn2 = (−1)n.
Discrete math about Fibonacci numbers

Determine if the set of functions is linearly independent:
1. f1(x)=cos2x, f2(x)=1, f3(x)=cos^2 x
2. f1(x)=e^ x, f2(x)=e^-x, f3(x)=senhx

(i) Suppose that f1, f2 are lower semicontinuous at a. Prove
that max{f1, f2} is lower semicontinuous at a.
(ii) Prove or disprove that f1 − f2 is lower semicontinuous
a.

Consider the following functions.
f1(x) = x, f2(x) = x-1, f3(x) = x+4
g(x) = c1f1(x) + c2f2(x) + c3f3(x)
Solve for c1, c2, and c3 so that g(x) = 0 on the interval (−∞, ∞).
If a nontrivial solution exists, state it. (If only the trivial
solution exists, enter the trivial solution {0, 0, 0}.)
{c1, c2, c3} =?
Determine whether f1, f2, f3 are linearly independent on the
interval (−∞, ∞).
linearly dependent or linearly independent?

The textile mill model -
Produces 3 types of fabric (F1, F2, F3) on 2 types of machines
(Regular, Special). Or Buy.
Time to delivery is 13 weeks.
15 regular machines (F2, F3) + 3 special machines (F1, F2, F3)
+ Buy (F1, F2, F3)
Demand for F1, F2, F3 is 45000, 76500, 10000 yards.
Special loom capacity in yards/hour – 4.7, 5.2, 4.4
Regular loom capacity in yards/hour – 0, 5.2, 4.4
Manufacture cost in $/yard – 0.65, 0.61,...

Let f1, f2, . . . ∈ k[x1, . . . , xn ] be an infinite collection
of polynomials and let I = < f1, f2, . . .> be the ideal they
generate. Prove that there is an integer N such that I = . Hint:
Use f1, f2, . . . to create an ascending chain of ideal

Consider the following functions.
f1(x) = x, f2(x) =
x2, f3(x) = 6x −
4x2
g(x) = c1f1(x) +
c2f2(x) + c3f3(x)
Solve for
c1, c2,
and
c3
so that
g(x) = 0
on the interval
(−∞, ∞).
If a nontrivial solution exists, state it. (If only the trivial
solution exists, enter the trivial solution {0, 0, 0}.)
{c1, c2, c3} =

The functions
f1(x) = x
and
f2(x) = x6
are orthogonal on
[−4, 4].
Find constants
C1
and
C2
such that
f3(x) = x + C1x2 +
C2x3
is orthogonal to both
f1
and
f2
on the same interval.

Three factories F1, F2 and F3 respectively produce 25%,
35% and 40% of the total number of electrical parts intended for
the assembly of a machine. These factories respectively produce 1%,
2% and 3% of defective parts.
We notice : The event A : "the part is produced by the F1
factory"
The event B : "the part is produced by the F2
factory"
The event C : "the part is produced by the F3 factory"
The event D : "the...

Let X be a set and let F1,F2 ⊆ P(X) be two σ-algebras on X. Let
G := {A ∩ B | A ∈ F1, B ∈ F2}. Prove the following statements:
(1) G is closed under finite intersections.
(2) σ(G) = σ(F1 ∪ F2).

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