Question and Answers Forum

Relation and FunctionsQuestion and Answers: Page 2

Question Number 196626 Answers: 0 Comments: 0

Σ_(x∈R) x = 0 and Π_(x∈R^∗ ) x = −1

$$\underset{{x}\in\mathbb{R}} {\sum}{x}\:=\:\mathrm{0}\:\:\:\:{and}\:\:\:\underset{{x}\in\mathbb{R}^{\ast} } {\prod}{x}\:=\:−\mathrm{1} \\ $$

Question Number 196621 Answers: 1 Comments: 0

find f(x) if f(x+(√(x^2 +1))) = (x/(x+1))

$$\mathrm{find}\:\:\mathrm{f}\left(\mathrm{x}\right)\:\:\mathrm{if} \\ $$$$\:\mathrm{f}\left(\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\right)\:=\:\frac{\mathrm{x}}{\mathrm{x}+\mathrm{1}} \\ $$

Question Number 196619 Answers: 1 Comments: 0

Question Number 196522 Answers: 1 Comments: 0

Prove that ∀n∈N ∫^( n+1) _( n) lnt dt ≤ ln(∫^( n+1) _n t dt)

$$\mathrm{Prove}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N} \\ $$$$\underset{\:\mathrm{n}} {\int}^{\:\mathrm{n}+\mathrm{1}} \mathrm{ln}{t}\:\mathrm{dt}\:\leqslant\:\mathrm{ln}\left(\underset{\mathrm{n}} {\int}^{\:\mathrm{n}+\mathrm{1}} {t}\:\mathrm{dt}\right) \\ $$

Question Number 196257 Answers: 1 Comments: 0

Question Number 196246 Answers: 1 Comments: 1

Question Number 196023 Answers: 1 Comments: 0

find the domain of definition of this function for t∈]0;1[ 𝛒(x)=∫_x ^(2x) (1/(lnt))dt ptiCantor

$${find}\:{the}\:{domain}\:{of}\:{definition}\:{of}\:{this} \\ $$$$\left.{function}\:{for}\:{t}\in\right]\mathrm{0};\mathrm{1}\left[\right. \\ $$$$\:\:\:\:\:\boldsymbol{\rho}\left({x}\right)=\int_{{x}} ^{\mathrm{2}{x}} \frac{\mathrm{1}}{{lnt}}{dt} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{ptiCantor} \\ $$

Question Number 195391 Answers: 2 Comments: 0

f^2 (x)+2f(x)=x^2 −8x+15 f(x)=?

$${f}^{\mathrm{2}} \left({x}\right)+\mathrm{2}{f}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{8}{x}+\mathrm{15} \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 194990 Answers: 0 Comments: 0

If (f(x))=x^2 −x+1 find the value of f(1971)+f(2021)+f(50)

$$\mathrm{I}{f}\:\left({f}\left({x}\right)\right)={x}^{\mathrm{2}} −{x}+\mathrm{1}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${f}\left(\mathrm{1971}\right)+{f}\left(\mathrm{2021}\right)+{f}\left(\mathrm{50}\right) \\ $$

Question Number 194896 Answers: 1 Comments: 0

$$\:\:\:\:\:\:\:\cancel{ } \\ $$

Question Number 198279 Answers: 1 Comments: 0

if f(x) is also differentiable on R such that f′(x) > f(x) ∀ x ∈ R and f(x_0 ) = 0 then prove that f(x) ≥ 0 ∀ x > x_0

$$\:\:\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{also}\:\mathrm{differentiable}\:\mathrm{on}\:\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\mathrm{f}'\left(\mathrm{x}\right)\:>\:\mathrm{f}\left(\mathrm{x}\right)\:\forall\:\mathrm{x}\:\in\:\mathbb{R}\:{and}\:\mathrm{f}\left(\mathrm{x}_{\mathrm{0}} \right)\:=\:\mathrm{0}\:\mathrm{then}\: \\ $$$$\:\:\mathrm{prove}\:\mathrm{that}\:\:\mathrm{f}\left(\mathrm{x}\right)\:\geqslant\:\mathrm{0}\:\forall\:\mathrm{x}\:>\:\mathrm{x}_{\mathrm{0}} \\ $$

Question Number 194688 Answers: 1 Comments: 0

find all function f: R → R such that ∀x, y∈R, f(x−f(y))=f(f(y))+xf(y)+f(x)−1.

$$\mathrm{find}\:\mathrm{all}\:\mathrm{function}\:{f}:\:\mathbb{R}\:\rightarrow\:\mathbb{R}\:\mathrm{such}\:\mathrm{that}\:\forall{x},\:{y}\in\mathbb{R}, \\ $$$${f}\left({x}−{f}\left({y}\right)\right)={f}\left({f}\left({y}\right)\right)+{xf}\left({y}\right)+{f}\left({x}\right)−\mathrm{1}. \\ $$

Question Number 193685 Answers: 2 Comments: 0

(f(x))^2 −4xf(x)+3=0 f^(−1) (3)=?

$$\:\:\:\:\:\:\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)^{\mathrm{2}} −\mathrm{4xf}\left(\mathrm{x}\right)+\mathrm{3}=\mathrm{0} \\ $$$$\:\:\:\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{3}\right)=?\: \\ $$

Question Number 193554 Answers: 2 Comments: 0

Question Number 193267 Answers: 1 Comments: 0

Question Number 192979 Answers: 0 Comments: 0

Can this be optimized (getting the minimum) using backprobagation? α(x_i ,y_i ,h_i )=(h_i −x_i )^2 +y_i ^2 β(y_i ,h_i )=y_i h_i ^2 −2uy_i h_i γ(h_i )=h_i ^4 −4uh_i ^3 +4u^2 h_i ^2 Cost=Σ_(i=0) ^m (cα(x_i ,y_i ,h_i )−2c^2 β(y_i ,h_i )+c^3 γ(h_i )) and how?

$$\mathrm{Can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{optimized}\:\left(\mathrm{getting}\:\mathrm{the}\:\mathrm{minimum}\right)\:\mathrm{using}\:\mathrm{backprobagation}? \\ $$$$ \\ $$$$\alpha\left({x}_{{i}} ,{y}_{{i}} ,{h}_{{i}} \right)=\left({h}_{{i}} −{x}_{{i}} \right)^{\mathrm{2}} +{y}_{{i}} ^{\mathrm{2}} \\ $$$$\beta\left({y}_{{i}} ,{h}_{{i}} \right)={y}_{{i}} {h}_{{i}} ^{\mathrm{2}} −\mathrm{2}{uy}_{{i}} {h}_{{i}} \\ $$$$\gamma\left({h}_{{i}} \right)={h}_{{i}} ^{\mathrm{4}} −\mathrm{4}{uh}_{{i}} ^{\mathrm{3}} +\mathrm{4}{u}^{\mathrm{2}} {h}_{{i}} ^{\mathrm{2}} \\ $$$$\mathrm{Cost}=\underset{{i}=\mathrm{0}} {\overset{{m}} {\sum}}\left(\mathrm{c}\alpha\left({x}_{{i}} ,{y}_{{i}} ,{h}_{{i}} \right)−\mathrm{2}{c}^{\mathrm{2}} \beta\left({y}_{{i}} ,{h}_{{i}} \right)+{c}^{\mathrm{3}} \gamma\left({h}_{{i}} \right)\right) \\ $$$$\mathrm{and}\:\mathrm{how}? \\ $$

Question Number 192181 Answers: 1 Comments: 0