For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima , inflection points, and asymptotic behavior. 297. y = x 3 + 4 x 2 + 3 x 3 x + 9
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima , inflection points, and asymptotic behavior. 297. y = x 3 + 4 x 2 + 3 x 3 x + 9
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.
297.
y
=
x
3
+
4
x
2
+
3
x
3
x
+
9
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
7. [10 marks]
Let G = (V,E) be a 3-connected graph with at least 6 vertices. Let C be a cycle in G
of length 5. We show how to find a longer cycle in G.
(a) Let x be a vertex of G that is not on C. Show that there are three C-paths
Po, P1, P2 that are disjoint except at the shared initial vertex and only intersect
C at their final vertices.
(b) Show that at least two of P0, P1, P2 have final vertices that are adjacent along C.
(c) Combine two of Po, P1, P2 with C to produce a cycle in G that is longer than C.
1. Let X and Y be random variables and suppose that A = F. Prove that
Z XI(A)+YI(A) is a random variable.
30. (a) What is meant by the term "product measur"?
AND
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