Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 127 Let us return to the problem. Because the limit is of the form 1∞, it is standard ...

128 3 Real Analysis 381.Does lim x→π/ 2 (sinx) cos^1 x exist? 382.For two positive integersmandn, compute lim x→ 0 √mcosx−√ncosx ...

3.2 Continuity, Derivatives, and Integrals 129 Summing up, we obtain f(x)−f (x 2 n ) =x ( 1 2 + 1 4 +···+ 1 2 n ) , which, whenn ...

130 3 Real Analysis The functionφis extended linearly over each open interval that was removed in the process of constructingC, ...

3.2 Continuity, Derivatives, and Integrals 131 390.Letaandbbe real numbers in the interval( 0 ,^12 )and letfbe a continuous real ...

132 3 Real Analysis g(x)assumes both positive and negative values on this interval. Being continuous,g has the intermediate valu ...

3.2 Continuity, Derivatives, and Integrals 133 the automaton remains inactive, producing the value 0. Because only the rightmost ...

134 3 Real Analysis 401.Prove that any convex polygonal surface can be divided by two perpendicular lines into four regions of e ...

3.2 Continuity, Derivatives, and Integrals 135 Solution.We prove that the function f(t)=at+bt+ct is increasing fort≥0. Its first ...

136 3 Real Analysis This follows by applying the product rule to the formula of the determinant. For our problem, consider the f ...

3.2 Continuity, Derivatives, and Integrals 137 404.For a nonzero real numberxprove thatex>x+1. 405.Find all positive real sol ...

138 3 Real Analysis orlimx→x 0 |f(x)|=limx→x 0 |g(x)|=∞, and if additionallylimx→x 0 f ′(x) g′(x)exists, then limx→x 0 f(x)g(x)e ...

3.2 Continuity, Derivatives, and Integrals 139 Rolle’s theorem.Iff:[a, b]→Ris continuous on[a, b], differentiable on(a, b), and ...

140 3 Real Analysis Example.Letf : R → Rbe a twice-differentiable function, with positive second derivative. Prove that f(x+f′(x ...

3.2 Continuity, Derivatives, and Integrals 141 Below you will find a variety of problems based on the above-mentioned theorems ( ...

142 3 Real Analysis 3.2.6 Convex Functions........................................ A function is called convex if any segment wi ...

3.2 Continuity, Derivatives, and Integrals 143 ∑n i= 1 xiyi≤ (n ∑ i= 1 xip ) 1 /p( n ∑ i= 1 yqi ) 1 /q , with equality if and on ...

144 3 Real Analysis Solution.A bounded convex function on( 0 ,∞)has a horizontal asymptote, so its deriva- tive tends to zero at ...

3.2 Continuity, Derivatives, and Integrals 145 432.Prove that if a sequence of positive real numbers(bn)nhas the property that(a ...

146 3 Real Analysis 436.Prove that for any natural numbern≥2 and any|x|≤1, ( 1 +x)n+( 1 −x)n≤ 2 n. 437.Prove that for any positi ...