Question and Answers Forum

AllQuestion and Answers: Page 604

Question Number 159121 Answers: 0 Comments: 0

Consider f(x) = x^3 + 2x −1. Use the intermidiate value theorem and the Rolle theorem to establish that the equation f(x) = 0 has a unique solution denoted a_0 ∈] 0,1[.

$$\mathrm{Consider} \\ $$$${f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{2}{x}\:−\mathrm{1}. \\ $$$$\mathrm{Use}\:\mathrm{the}\:\mathrm{intermidiate}\:\mathrm{value}\:\mathrm{theorem}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{Rolle}\:\mathrm{theorem}\:\mathrm{to}\:\mathrm{establish}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution} \\ $$$$\left.\mathrm{denoted}\:{a}_{\mathrm{0}} \in\right]\:\mathrm{0},\mathrm{1}\left[.\:\right. \\ $$

Question Number 159119 Answers: 0 Comments: 0

Assume x;y;z>0 and x^2 +y^2 +z^2 =12 Prove that: Σ_(cycl) (((x/y) + 1 + (y/x))/((1/x) + (1/y))) ≤ 9

$$\mathrm{Assume}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{12} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\underset{\boldsymbol{\mathrm{cycl}}} {\sum}\:\frac{\frac{\mathrm{x}}{\mathrm{y}}\:+\:\mathrm{1}\:+\:\frac{\mathrm{y}}{\mathrm{x}}}{\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}}}\:\leqslant\:\mathrm{9} \\ $$

Question Number 159118 Answers: 0 Comments: 0

Let F(r) = { ((mkr, 0 ≤ r < R)),((((mgR^2 )/r^2 ), r≥ R)) :} find D_F

$$\mathrm{Let}\:{F}\left({r}\right)\:=\:\begin{cases}{{mkr},\:\:\mathrm{0}\:\leqslant\:{r}\:<\:{R}}\\{\frac{{m}\mathrm{g}{R}^{\mathrm{2}} }{{r}^{\mathrm{2}} },\:{r}\geqslant\:{R}}\end{cases} \\ $$$$\mathrm{find}\:{D}_{{F}} \\ $$

Question Number 159111 Answers: 1 Comments: 0

Given A^→ =5t^2 i^→ +tj^→ −t^3 k^→ and B^→ =sin(t)i^→ −cos(t)j^→ . Calculate ((d(A^→ .B^→ ))/dx) ; ((d(A^→ ∧B^→ ))/dx) and ((d(A^→ .A^→ ))/dx).

Question Number 159109 Answers: 0 Comments: 0

Show that ∀ V_i ^→ and V_j ^→ : V_i ^→ ∧[V_j ^→ ∧(V_j ^→ ∧V_i ^→ )]=−V_j ^→ ∧[V_i ^→ ∧(V_i ^→ ∧V_j ^→ )]

$${Show}\:{that}\:\forall\:\overset{\rightarrow} {{V}}_{{i}} \:{and}\:\overset{\rightarrow} {{V}}_{{j}} : \\ $$$$\overset{\rightarrow} {{V}}_{{i}} \wedge\left[\overset{\rightarrow} {{V}}_{{j}} \wedge\left(\overset{\rightarrow} {{V}}_{{j}} \wedge\overset{\rightarrow} {{V}}_{{i}} \right)\right]=−\overset{\rightarrow} {{V}}_{{j}} \wedge\left[\overset{\rightarrow} {{V}}_{{i}} \wedge\left(\overset{\rightarrow} {{V}}_{{i}} \wedge\overset{\rightarrow} {{V}}_{{j}} \right)\right] \\ $$

Question Number 160090 Answers: 0 Comments: 2

There are 40 oranges, 20 apples, and 20 lemons in a bag. What is the minimum number of fruits that you have to take out of the bag with your eyes closed before you are sure that one of them is an orange?

$$\mathrm{There}\:\mathrm{are}\:\mathrm{40}\:\mathrm{oranges},\:\mathrm{20}\:\mathrm{apples},\:\mathrm{and}\: \\ $$$$\mathrm{20}\:\mathrm{lemons}\:\mathrm{in}\:\mathrm{a}\:\mathrm{bag}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{minimum}\:\mathrm{number}\:\mathrm{of}\:\mathrm{fruits}\:\mathrm{that}\:\mathrm{you}\: \\ $$$$\mathrm{have}\:\mathrm{to}\:\mathrm{take}\:\mathrm{out}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bag}\:\mathrm{with}\:\mathrm{your}\: \\ $$$$\mathrm{eyes}\:\mathrm{closed}\:\mathrm{before}\:\mathrm{you}\:\mathrm{are}\:\mathrm{sure}\:\mathrm{that}\: \\ $$$$\mathrm{one}\:\mathrm{of}\:\mathrm{them}\:\mathrm{is}\:\mathrm{an}\:\mathrm{orange}? \\ $$

Question Number 159125 Answers: 0 Comments: 3

for a, b, c >0 and a+b+c=2 find min(2ab^2 +b^3 c, 2bc^2 +c^3 a, 2ca^2 +a^3 b) or disprove that such a minimum doesn′t exist.

$${for}\:{a},\:{b},\:{c}\:>\mathrm{0}\:{and}\:{a}+{b}+{c}=\mathrm{2} \\ $$$${find}\:{min}\left(\mathrm{2}{ab}^{\mathrm{2}} +{b}^{\mathrm{3}} {c},\:\mathrm{2}{bc}^{\mathrm{2}} +{c}^{\mathrm{3}} {a},\:\mathrm{2}{ca}^{\mathrm{2}} +{a}^{\mathrm{3}} {b}\right) \\ $$$${or}\:{disprove}\:{that}\:{such}\:{a}\:{minimum} \\ $$$${doesn}'{t}\:{exist}. \\ $$

Question Number 159106 Answers: 0 Comments: 0

∫_0 ^π ((sin(nz))/(z^4 sin(πz)))dz

$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{sin}\left({nz}\right)}{\mathrm{z}^{\mathrm{4}} \mathrm{sin}\left(\pi{z}\right)}{dz} \\ $$

Question Number 159099 Answers: 1 Comments: 0