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Question Number 2675    Answers: 1   Comments: 8

Bases on suggestion from Filup and some discussion on that I am suggesting that we sequence, series and related function as a topic for this month. ζ(x)=Σ_(n=1) ^∞ n^(−x) , x∈R, x>1 Show that ζ(x)=(1/(Γ(x)))∫_0 ^∞ (t^(x−1) /(e^t −1))dt

$$\mathrm{Bases}\:\mathrm{on}\:\mathrm{suggestion}\:\mathrm{from}\:\mathrm{Filup}\:\mathrm{and}\:\mathrm{some} \\ $$$$\mathrm{discussion}\:\mathrm{on}\:\mathrm{that}\:\mathrm{I}\:\mathrm{am}\:\mathrm{suggesting}\:\mathrm{that}\:\mathrm{we} \\ $$$$\mathrm{sequence},\:\mathrm{series}\:\mathrm{and}\:\mathrm{related}\:\mathrm{function}\:\mathrm{as}\:\mathrm{a} \\ $$$$\mathrm{topic}\:\mathrm{for}\:\mathrm{this}\:\mathrm{month}. \\ $$$$\zeta\left({x}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{n}^{−{x}} ,\:{x}\in\mathbb{R},\:{x}>\mathrm{1} \\ $$$$\mathrm{Show}\:\mathrm{that} \\ $$$$\zeta\left({x}\right)=\frac{\mathrm{1}}{\Gamma\left({x}\right)}\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{x}−\mathrm{1}} }{{e}^{{t}} −\mathrm{1}}\mathrm{d}{t} \\ $$

Question Number 2672    Answers: 1   Comments: 2

I have 4 collinear points A(a,0), B(b,0), C(c,0) and D(d,0) where ∀a,b,c,d>0. Find a point E(x,y) such that the following expression is minimised: 2(AE+BE+CE+DE).

$${I}\:{have}\:\mathrm{4}\:{collinear}\:{points}\:{A}\left({a},\mathrm{0}\right), \\ $$$${B}\left({b},\mathrm{0}\right),\:{C}\left({c},\mathrm{0}\right)\:{and}\:{D}\left({d},\mathrm{0}\right)\:{where}\: \\ $$$$\forall{a},{b},{c},{d}>\mathrm{0}.\:{Find}\:{a}\:{point}\:{E}\left({x},{y}\right)\:{such} \\ $$$${that}\:{the}\:{following}\:{expression}\:{is} \\ $$$${minimised}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\left({AE}+{BE}+{CE}+{DE}\right). \\ $$

Question Number 2655    Answers: 2   Comments: 0

Prove by contradiction that there are no whole number solutions (x,y,z) to the equation z^2 =x^2 +y^2 where both x and y are odd.

$${Prove}\:{by}\:{contradiction}\:{that}\:{there} \\ $$$${are}\:{no}\:{whole}\:{number}\:{solutions}\:\left({x},{y},{z}\right) \\ $$$${to}\:{the}\:{equation}\:{z}^{\mathrm{2}} ={x}^{\mathrm{2}} +{y}^{\mathrm{2}} \\ $$$${where}\:{both}\:{x}\:{and}\:{y}\:{are}\:{odd}. \\ $$

Question Number 2649    Answers: 0   Comments: 10

Ok guys. We all LOVE mathematics. So, I have came up with a simple and possibly effect way to not only improve our abilities as mathematicians, but also have some fun, too. We should hold a competition every now and again. We can, for example have questions such as the following: (v^→ , k^→ )∈R^n , v^→ =⟨v_1 , ..., v_n ⟩, k^→ =⟨1, 1, ..., 1⟩. Prove that if v•k=0, v and k are orthoganal. Do so in the most creative way(s) or most complex. Let this post be a trial competition. If this goes well, we should continue doing more in the future! Good Luck Edit: Let me know what you all think!

$$\mathrm{Ok}\:\mathrm{guys}.\:\mathrm{We}\:\mathrm{all}\:\mathscr{LOVE}\:\mathrm{mathematics}. \\ $$$$\mathrm{So},\:\mathrm{I}\:\mathrm{have}\:\mathrm{came}\:\mathrm{up}\:\mathrm{with}\:\mathrm{a}\:\mathrm{simple}\:\mathrm{and} \\ $$$$\mathrm{possibly}\:\mathrm{effect}\:\mathrm{way}\:\mathrm{to}\:\mathrm{not}\:\mathrm{only}\:\mathrm{improve} \\ $$$$\mathrm{our}\:\mathrm{abilities}\:\mathrm{as}\:\mathrm{mathematicians},\:\mathrm{but}\:\mathrm{also} \\ $$$$\mathrm{have}\:\mathrm{some}\:\mathrm{fun},\:\mathrm{too}. \\ $$$$ \\ $$$$\mathrm{We}\:\mathrm{should}\:\mathrm{hold}\:\mathrm{a}\:\mathrm{competition}\:\mathrm{every} \\ $$$$\mathrm{now}\:\mathrm{and}\:\mathrm{again}.\:\mathrm{We}\:\mathrm{can},\:\mathrm{for}\:\mathrm{example} \\ $$$$\mathrm{have}\:\mathrm{questions}\:\mathrm{such}\:\mathrm{as}\:\mathrm{the}\:\mathrm{following}: \\ $$$$ \\ $$$$\left(\overset{\rightarrow} {\boldsymbol{{v}}},\:\overset{\rightarrow} {\boldsymbol{{k}}}\right)\in\mathbb{R}^{{n}} ,\:\overset{\rightarrow} {\boldsymbol{{v}}}=\langle{v}_{\mathrm{1}} ,\:...,\:{v}_{{n}} \rangle,\:\overset{\rightarrow} {\boldsymbol{{k}}}=\langle\mathrm{1},\:\mathrm{1},\:...,\:\mathrm{1}\rangle. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\boldsymbol{{v}}\bullet\boldsymbol{{k}}=\mathrm{0},\:\boldsymbol{{v}}\:\mathrm{and}\:\boldsymbol{{k}}\:\mathrm{are}\:\mathrm{orthoganal}. \\ $$$$\mathrm{Do}\:\mathrm{so}\:\mathrm{in}\:\mathrm{the}\:\mathrm{most}\:\mathrm{creative}\:\mathrm{way}\left(\mathrm{s}\right)\:\mathrm{or}\:\mathrm{most} \\ $$$$\mathrm{complex}. \\ $$$$ \\ $$$$\mathrm{Let}\:\mathrm{this}\:\mathrm{post}\:\mathrm{be}\:\mathrm{a}\:{trial}\:\mathrm{competition}. \\ $$$$\mathrm{If}\:\mathrm{this}\:\mathrm{goes}\:\mathrm{well},\:\mathrm{we}\:\mathrm{should}\:\mathrm{continue} \\ $$$$\mathrm{doing}\:\mathrm{more}\:\mathrm{in}\:\mathrm{the}\:\mathrm{future}! \\ $$$$ \\ $$$${Good}\:{Luck} \\ $$$$ \\ $$$${Edit}: \\ $$$$\mathrm{Let}\:\mathrm{me}\:\mathrm{know}\:\mathrm{what}\:\mathrm{you}\:\mathrm{all}\:\mathrm{think}! \\ $$

Question Number 2645    Answers: 1   Comments: 0

A=∫_N_1 ^N_2 ⌊x⌋dx (N_1 , N_2 )∈Z, N_1 <N_2 Solve for A

$${A}=\int_{{N}_{\mathrm{1}} } ^{{N}_{\mathrm{2}} } \lfloor{x}\rfloor{dx} \\ $$$$\left({N}_{\mathrm{1}} ,\:{N}_{\mathrm{2}} \right)\in\mathbb{Z},\:\:\:{N}_{\mathrm{1}} <{N}_{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{Solve}\:\mathrm{for}\:{A} \\ $$

Question Number 2644    Answers: 1   Comments: 1

a_(n+1) =(a_n /n)+n a_1 =1 a_n =??? n∈N^∗

$${a}_{{n}+\mathrm{1}} =\frac{{a}_{{n}} }{{n}}+{n} \\ $$$${a}_{\mathrm{1}} =\mathrm{1} \\ $$$${a}_{{n}} =???\:{n}\in\mathbb{N}^{\ast} \\ $$

Question Number 2642    Answers: 1   Comments: 1

n lines are drawn inside a circle in such a way that the circle has been divided in maximum number of parts. Determine this maximum number.

$${n}\:{lines}\:{are}\:{drawn}\:{inside}\:{a}\:{circle}\:{in}\:{such}\:{a}\:{way}\:{that}\: \\ $$$${the}\:{circle}\:{has}\:{been}\:{divided}\:{in}\:{maximum}\:{number}\:{of} \\ $$$${parts}.\:{Determine}\:{this}\:{maximum}\:{number}. \\ $$

Question Number 2629    Answers: 1   Comments: 2

Can you evalate: A=∫_a ^( b) ⌊f(x)⌋dx for example: ∫_(0.5) ^( 2.5) ⌊x^2 ⌋dx

$$\mathrm{Can}\:\mathrm{you}\:\mathrm{evalate}: \\ $$$${A}=\int_{{a}} ^{\:{b}} \lfloor{f}\left({x}\right)\rfloor{dx} \\ $$$$ \\ $$$${for}\:{example}: \\ $$$$\int_{\mathrm{0}.\mathrm{5}} ^{\:\mathrm{2}.\mathrm{5}} \lfloor{x}^{\mathrm{2}} \rfloor{dx} \\ $$

Question Number 2624    Answers: 2   Comments: 0

Old question related to greatest int function. lim_(x→0) ⌊1+x⌋=1 ⌊1⌋=1 lim_(x→0) ⌊1−x⌋=?

$$\mathrm{Old}\:\mathrm{question}\:\mathrm{related}\:\mathrm{to}\:\mathrm{greatest}\:\mathrm{int}\:\mathrm{function}. \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\lfloor\mathrm{1}+{x}\rfloor=\mathrm{1} \\ $$$$\lfloor\mathrm{1}\rfloor=\mathrm{1} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\lfloor\mathrm{1}−{x}\rfloor=? \\ $$

Question Number 2619    Answers: 2   Comments: 1

The sums of the first n terms of two AP ′s are in the ratio 3n+31 : 5n−3 . Show that their 9^(th) terms are equal.

$${The}\:{sums}\:{of}\:{the}\:{first}\:\:{n}\:\:\:{terms}\:{of}\:{two}\:{AP}\:'{s}\:{are} \\ $$$${in}\:{the}\:{ratio}\:\:\mathrm{3}{n}+\mathrm{31}\::\:\:\mathrm{5}{n}−\mathrm{3}\:.\:{Show}\:{that}\:{their}\:\mathrm{9}^{{th}} \:{terms} \\ $$$${are}\:{equal}. \\ $$

Question Number 2621    Answers: 1   Comments: 0

If you have n non−parallel lines in a plane, how many points of intersection are there?

$${If}\:{you}\:{have}\:{n}\:{non}−{parallel} \\ $$$${lines}\:{in}\:{a}\:{plane},\:{how}\:{many}\:{points} \\ $$$${of}\:{intersection}\:{are}\:{there}? \\ $$

Question Number 2604    Answers: 1   Comments: 0

f(x,y)=f(x−1,y−x)+1 f(x,y)=ye^x ,x≤0 f(5,6)=?

$${f}\left({x},{y}\right)={f}\left({x}−\mathrm{1},{y}−{x}\right)+\mathrm{1} \\ $$$${f}\left({x},{y}\right)={ye}^{{x}} ,{x}\leqslant\mathrm{0} \\ $$$${f}\left(\mathrm{5},\mathrm{6}\right)=? \\ $$

Question Number 2603    Answers: 2   Comments: 0

You have a 3 litre jug and a 5 litre jug. Make 4 litres.

$${You}\:{have}\:{a}\:\mathrm{3}\:{litre}\:{jug}\:{and}\:{a}\:\mathrm{5}\:{litre}\:{jug}.\:{Make}\:\mathrm{4}\:{litres}. \\ $$

Question Number 2602    Answers: 1   Comments: 0

Solve the following d.e for v in terms of s c−kv=v(dv/ds).

$${Solve}\:{the}\:{following}\:{d}.{e}\:{for}\:{v}\:{in}\:{terms}\:{of}\:{s} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{c}−{kv}={v}\frac{{dv}}{{ds}}. \\ $$$$ \\ $$

Question Number 2589    Answers: 2   Comments: 3

f_n =(1/n)(n+f_(n−1) ) f_1 =1 Evaluate: S=Σ_(i=1) ^m f_i

$${f}_{{n}} =\frac{\mathrm{1}}{{n}}\left({n}+{f}_{{n}−\mathrm{1}} \right) \\ $$$${f}_{\mathrm{1}} =\mathrm{1} \\ $$$$ \\ $$$$\mathrm{Evaluate}: \\ $$$${S}=\underset{{i}=\mathrm{1}} {\overset{{m}} {\sum}}{f}_{{i}} \\ $$

Question Number 2720    Answers: 1   Comments: 1

Analytical Continuation Sum of the below divergent series was shown to be using analytical continuation. Σ_(i=1) ^n 2^(i−1) =−1 ...(A) ζ(−1)=Σ_(i=0) ^∞ i=−(1/(12)) ...(B) While reading about analytical continuation, I found the found the following: If f_1 is a analytic function over domain D_1 and If f_2 is a analytic function over domain D_2 and f_1 =f_2 on D_1 ∩D_2 , f_2 is called analytical continuation of f_1 and vice versa. Question: In case of series (A) and (B) above what function is used as f_2 to find the sum also what is used as f_1 ?

$$\mathrm{Analytical}\:\mathrm{Continuation} \\ $$$$\mathrm{Sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{below}\:\mathrm{divergent}\:\mathrm{series}\:\mathrm{was} \\ $$$$\mathrm{shown}\:\mathrm{to}\:\mathrm{be}\:\mathrm{using}\:\mathrm{analytical}\:\mathrm{continuation}. \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{2}^{{i}−\mathrm{1}} =−\mathrm{1}\:\:\:\:\:\:...\left(\mathrm{A}\right) \\ $$$$\zeta\left(−\mathrm{1}\right)=\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}{i}=−\frac{\mathrm{1}}{\mathrm{12}}\:\:\:\:\:\:...\left(\mathrm{B}\right) \\ $$$$\mathrm{While}\:\mathrm{reading}\:\mathrm{about}\:\mathrm{analytical}\:\mathrm{continuation}, \\ $$$$\mathrm{I}\:\mathrm{found}\:\mathrm{the}\:\mathrm{found}\:\mathrm{the}\:\mathrm{following}: \\ $$$$\mathrm{If}\:{f}_{\mathrm{1}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{analytic}\:\mathrm{function}\:\mathrm{over}\:\mathrm{domain}\:\mathrm{D}_{\mathrm{1}} \:\mathrm{and} \\ $$$$\mathrm{If}\:{f}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{analytic}\:\mathrm{function}\:\mathrm{over}\:\mathrm{domain}\:\mathrm{D}_{\mathrm{2}} \:\mathrm{and} \\ $$$${f}_{\mathrm{1}} ={f}_{\mathrm{2}} \:\mathrm{on}\:\mathrm{D}_{\mathrm{1}} \cap\mathrm{D}_{\mathrm{2}} ,\:{f}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{called}\:\mathrm{analytical}\: \\ $$$$\mathrm{continuation}\:\mathrm{of}\:{f}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{vice}\:\mathrm{versa}. \\ $$$$\boldsymbol{\mathrm{Question}}: \\ $$$$\mathrm{In}\:\mathrm{case}\:\mathrm{of}\:\mathrm{series}\:\left(\mathrm{A}\right)\:\mathrm{and}\:\left(\mathrm{B}\right)\:\mathrm{above}\:\mathrm{what}\:\mathrm{function} \\ $$$$\mathrm{is}\:\mathrm{used}\:\mathrm{as}\:{f}_{\mathrm{2}} \:\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{also}\:\mathrm{what}\:\mathrm{is}\:\mathrm{used}\:\mathrm{as}\:{f}_{\mathrm{1}} ? \\ $$

Question Number 2575    Answers: 2   Comments: 4

a_1 =0 a_n =27×a_(n−1) +(n−1) Σ_(k=1) ^m a_k =?

$${a}_{\mathrm{1}} =\mathrm{0} \\ $$$${a}_{{n}} =\mathrm{27}×{a}_{{n}−\mathrm{1}} +\left({n}−\mathrm{1}\right) \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{m}} {\sum}}{a}_{{k}} =? \\ $$

Question Number 2564    Answers: 1   Comments: 0

A= [(1,2),(2,5) ],B= [(3,(−1)),(4,(+2)) ] determine A^B

$${A}=\begin{bmatrix}{\mathrm{1}}&{\mathrm{2}}\\{\mathrm{2}}&{\mathrm{5}}\end{bmatrix},{B}=\begin{bmatrix}{\mathrm{3}}&{−\mathrm{1}}\\{\mathrm{4}}&{+\mathrm{2}}\end{bmatrix} \\ $$$${determine}\:{A}^{{B}} \\ $$

Question Number 2562    Answers: 0   Comments: 1

App Updates: We have updated the app with below changes: • ad banner removed from all screens This provides larger screen area for equation writing and avoids distraction. • confirmation on delete posts on Q and A forum

$$\mathrm{App}\:\mathrm{Updates}: \\ $$$$\mathrm{We}\:\mathrm{have}\:\mathrm{updated}\:\mathrm{the}\:\mathrm{app}\:\mathrm{with}\:\mathrm{below}\:\mathrm{changes}: \\ $$$$\bullet\:\boldsymbol{\mathrm{ad}}\:\boldsymbol{\mathrm{banner}}\:\boldsymbol{\mathrm{removed}}\:\boldsymbol{\mathrm{from}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{screens}} \\ $$$$\:\:\:\:\mathrm{This}\:\mathrm{provides}\:\mathrm{larger}\:\mathrm{screen}\:\mathrm{area}\:\mathrm{for} \\ $$$$\:\:\:\:\:\mathrm{equation}\:\mathrm{writing}\:\mathrm{and}\:\mathrm{avoids}\:\mathrm{distraction}. \\ $$$$\bullet\:\boldsymbol{\mathrm{confirmation}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{delete}}\:\boldsymbol{\mathrm{posts}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{Q}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{A}}\:\boldsymbol{\mathrm{forum}} \\ $$

Question Number 2548    Answers: 1   Comments: 0

For a function y=f(x), inflection points/stationary points are when (df/dx)=0. For a function z=f(x, y), can you find these points through a similar method? Is it something like (∂f/∂x)=0 and (∂f/∂y)=0?

$$\mathrm{For}\:\mathrm{a}\:\mathrm{function}\:{y}={f}\left({x}\right), \\ $$$$\mathrm{inflection}\:\mathrm{points}/\mathrm{stationary}\:\mathrm{points}\:\mathrm{are} \\ $$$$\mathrm{when}\:\:\frac{{df}}{{dx}}=\mathrm{0}. \\ $$$$ \\ $$$$\mathrm{For}\:\mathrm{a}\:\mathrm{function}\:{z}={f}\left({x},\:{y}\right),\:\mathrm{can}\:\mathrm{you}\:\mathrm{find} \\ $$$$\mathrm{these}\:\mathrm{points}\:\mathrm{through}\:\mathrm{a}\:\mathrm{similar}\:\mathrm{method}? \\ $$$$ \\ $$$$\mathrm{Is}\:\mathrm{it}\:\mathrm{something}\:\mathrm{like}\:\frac{\partial{f}}{\partial{x}}=\mathrm{0}\:\mathrm{and}\:\frac{\partial{f}}{\partial{y}}=\mathrm{0}? \\ $$

Question Number 2546    Answers: 0   Comments: 4

If A and B are two matrices of suitable order does there exist definition for A^B ?

$$\boldsymbol{\mathrm{If}}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{two}\:\mathrm{matrices}\:\mathrm{of}\:\mathrm{suitable}\:\mathrm{order} \\ $$$$\mathrm{does}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{definition}\:\mathrm{for}\:\mathrm{A}^{\mathrm{B}} \:? \\ $$

Question Number 2545    Answers: 2   Comments: 1

Prove that 3^(3n) −26n−1 is divisible by 676, where n∈Z^+

$$\mathcal{P}{rove}\:{that}\:\mathrm{3}^{\mathrm{3}{n}} −\mathrm{26}{n}−\mathrm{1}\:{is}\:{divisible}\:{by}\:\mathrm{676}, \\ $$$${where}\:{n}\in\mathbb{Z}^{+} \\ $$

Question Number 2544    Answers: 1   Comments: 0

Can you Generalize the following? 1+2+3+...+n=(1/2)(n)(n+1) 1^2 +2^2 +3^2 +...+n^2 =(1/6)(n)(n+1)(2n+1 1^3 +2^3 +3^3 +...+n^3 =[(1/2)(n)(n+1)]^2 .... ..... ..... ... ..... ..... ...... ... ... .... .... ... ... 1^k +2^k +3^k +...+n^k =???

$$\mathcal{C}{an}\:{you}\:\mathcal{G}{eneralize}\:{the}\:{following}? \\ $$$$\mathrm{1}+\mathrm{2}+\mathrm{3}+...+{n}=\frac{\mathrm{1}}{\mathrm{2}}\left({n}\right)\left({n}+\mathrm{1}\right) \\ $$$$\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +...+{n}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{6}}\left({n}\right)\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right. \\ $$$$\mathrm{1}^{\mathrm{3}} +\mathrm{2}^{\mathrm{3}} +\mathrm{3}^{\mathrm{3}} +...+{n}^{\mathrm{3}} =\left[\frac{\mathrm{1}}{\mathrm{2}}\left({n}\right)\left({n}+\mathrm{1}\right)\right]^{\mathrm{2}} \\ $$$$....\:\:\:.....\:\:\:\:.....\:\:...\:.....\:.....\:...... \\ $$$$...\:\:...\:\:\:....\:....\:\:\:\:\:\:...\:\:... \\ $$$$\mathrm{1}^{{k}} +\mathrm{2}^{{k}} +\mathrm{3}^{{k}} +...+{n}^{{k}} =??? \\ $$

Question Number 2526    Answers: 1   Comments: 2

Evaluate: ∫_(−∞) ^∞ e^(−x^2 ) dx Please show and explain working

$$\mathrm{Evaluate}:\:\underset{−\infty} {\overset{\infty} {\int}}{e}^{−{x}^{\mathrm{2}} } {dx} \\ $$$$\mathrm{Please}\:\mathrm{show}\:\mathrm{and}\:\mathrm{explain}\:\mathrm{working} \\ $$

Question Number 2524    Answers: 1   Comments: 0

f:R^2 →R f(x,y)=x^2 −y^2 y≥0 f(x,y)=x^2 +y^2 y≤0 find (x,y) for min f(x,y) (x,y) for f(x,y)=1

$${f}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R} \\ $$$${f}\left({x},{y}\right)={x}^{\mathrm{2}} −{y}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:{y}\geqslant\mathrm{0} \\ $$$${f}\left({x},{y}\right)={x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:{y}\leqslant\mathrm{0} \\ $$$$\mathrm{find} \\ $$$$\left({x},{y}\right)\:\mathrm{for}\:\mathrm{min}\:{f}\left({x},{y}\right) \\ $$$$\left({x},{y}\right)\:\mathrm{for}\:{f}\left({x},{y}\right)=\mathrm{1} \\ $$$$ \\ $$

Question Number 2504    Answers: 1   Comments: 0

f(x,y)=f(x,y−x) f(x,y)=f(y,x) f(0,y)=y^2 does g(x):=f(x,x) g(−x)=^? g(x) f(10,5)=?

$${f}\left({x},{y}\right)={f}\left({x},{y}−{x}\right) \\ $$$${f}\left({x},{y}\right)={f}\left({y},{x}\right) \\ $$$${f}\left(\mathrm{0},{y}\right)={y}^{\mathrm{2}} \\ $$$$\mathrm{does} \\ $$$${g}\left({x}\right):={f}\left({x},{x}\right) \\ $$$${g}\left(−{x}\right)\overset{?} {=}{g}\left({x}\right) \\ $$$${f}\left(\mathrm{10},\mathrm{5}\right)=? \\ $$

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