Cauchy–Schwarz Inequality The definition u · v = | u | | v | cos θ implies that | u · v | ≤ | u | | v | ( because | cos θ | ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and has many consequences. 88. Algebra inequality Show that ( u 1 + u 2 + u 3 ) 2 ≤ 3 ( u 1 2 + u 2 2 + u 3 2 ) , for any real numbers u 1 , u 2 , and u 3 . ( Hint: Use the Cauchy–Schwarz Inequality in three dimensions with u = 〈 u 1 , u 2 , u 3 〉 and choose v in the right way.)
Cauchy–Schwarz Inequality The definition u · v = | u | | v | cos θ implies that | u · v | ≤ | u | | v | ( because | cos θ | ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and has many consequences. 88. Algebra inequality Show that ( u 1 + u 2 + u 3 ) 2 ≤ 3 ( u 1 2 + u 2 2 + u 3 2 ) , for any real numbers u 1 , u 2 , and u 3 . ( Hint: Use the Cauchy–Schwarz Inequality in three dimensions with u = 〈 u 1 , u 2 , u 3 〉 and choose v in the right way.)
Cauchy–Schwarz InequalityThe definitionu · v = |u| |v| cos θ implies that |u · v| ≤ | u| |v| (because | cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and has many consequences.
88. Algebra inequality Show that
(
u
1
+
u
2
+
u
3
)
2
≤
3
(
u
1
2
+
u
2
2
+
u
3
2
)
,
for any real numbers u1, u2, and u3. (Hint: Use the Cauchy–Schwarz Inequality in three dimensions with u = 〈u1, u2, u3〉 and choose v in the right way.)
Calculus I
In the exercise f(x)= cos x + sin x; [0,2pi], find the following
1.) Search for critical points2.) Search if it grows or decreases3.) Search for local maximum and minimum
Using the First Derivative Test proved in the videos, prove the following version of the First Derivative Test: If f′ is continuous on the interval [a,b] and if f has exactly one critical point c then f has a maximum at c if f′(a′)>0 and f′(b′)<0 for some a′ and b′ such that a<a′<c<b′<b.
(Triangle inequality) ∀x, y ∈ R : |x + y| ≤ |x| + |y|. Proof. Exercise. Use that |x| = max{x, −x}.
University Calculus: Early Transcendentals (4th Edition)
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