Question and Answers Forum

Why are following statements wrong? a) There exists a function with domain R satisfying f(x)<0 ∀x , f′(x)>0∀x and f′′(x)>0∀x. b) If f′′(c)=0 then (c,f(c)) is an inflection point.

$$\mathrm{Why}\:\mathrm{are}\:\mathrm{following}\:\mathrm{statements}\:\mathrm{wrong}? \\ $$$$\left.\mathrm{a}\right)\:\mathrm{There}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{function}\:\mathrm{with}\:\mathrm{domain}\: \\ $$$$\mathrm{R}\:\mathrm{satisfying}\:\mathrm{f}\left(\mathrm{x}\right)<\mathrm{0}\:\forall\mathrm{x}\:,\:\mathrm{f}'\left(\mathrm{x}\right)>\mathrm{0}\forall\mathrm{x}\:\mathrm{and} \\ $$$$\mathrm{f}''\left(\mathrm{x}\right)>\mathrm{0}\forall\mathrm{x}. \\ $$$$ \\ $$$$\left.\mathrm{b}\right)\:\mathrm{If}\:\mathrm{f}''\left(\mathrm{c}\right)=\mathrm{0}\:\mathrm{then}\:\left(\mathrm{c},\mathrm{f}\left(\mathrm{c}\right)\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{inflection} \\ $$$$\mathrm{point}. \\ $$

Question Number 37108 Answers: 1 Comments: 0

If 4x+8cos x+tan x−2sec x−4log {cosx(1+sin x)}≥6 ∀ x ε [0,ψ) then largest value of ψ is ?

$$\mathrm{If}\:\mathrm{4}{x}+\mathrm{8cos}\:{x}+\mathrm{tan}\:{x}−\mathrm{2sec}\:{x}−\mathrm{4log}\:\left\{\mathrm{cos}{x}\left(\mathrm{1}+\mathrm{sin}\:{x}\right)\right\}\geqslant\mathrm{6} \\ $$$$\forall\:{x}\:\epsilon\:\left[\mathrm{0},\psi\right)\:\mathrm{then}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:\psi\:\mathrm{is}\:? \\ $$

Question Number 37079 Answers: 1 Comments: 0

Question Number 37071 Answers: 2 Comments: 1

find the value of ∫_0 ^(π/2) ((xdx)/(1+cosx))

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{xdx}}{\mathrm{1}+{cosx}} \\ $$

Question Number 37067 Answers: 2 Comments: 1

find ∫ (dx/((x+1)(√(1+x^2 ))))

$${find}\:\int\:\:\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }} \\ $$

Question Number 37036 Answers: 0 Comments: 3

∫_0 ^( a) (1−((b−x)/(√((b−x)^2 +cx)))) dx = ?

$$\int_{\mathrm{0}} ^{\:\:{a}} \left(\mathrm{1}−\frac{{b}−{x}}{\sqrt{\left({b}−{x}\right)^{\mathrm{2}} +{cx}}}\right)\:{dx}\:=\:? \\ $$

Question Number 37028 Answers: 1 Comments: 0

∫((2x−1)/(5x^2 −x+2)) dx = ?

$$\int\frac{\mathrm{2}{x}−\mathrm{1}}{\mathrm{5}{x}^{\mathrm{2}} −{x}+\mathrm{2}}\:{dx}\:\:=\:\:? \\ $$

Question Number 37018 Answers: 3 Comments: 3

Question Number 37004 Answers: 3 Comments: 0

Question Number 36997 Answers: 2 Comments: 1

∫ (√(((1+x)/x) ))dx = ?

$$\int\:\sqrt{\frac{\mathrm{1}+{x}}{{x}}\:}{dx}\:=\:? \\ $$

Question Number 36965 Answers: 2 Comments: 3

∫((x^5 −x^4 +x^3 −1)/((x^2 −x+1)^3 ))dx=

$$\int\frac{{x}^{\mathrm{5}} −{x}^{\mathrm{4}} +{x}^{\mathrm{3}} −\mathrm{1}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}= \\ $$

Question Number 36957 Answers: 2 Comments: 0

Interval in which given function is decreasing. f(x)= (2^x −1)(2^x −2)^2

$$\mathrm{Interval}\:\mathrm{in}\:\mathrm{which}\:\mathrm{given}\:\mathrm{function}\:\mathrm{is} \\ $$$${decreasing}. \\ $$$$\mathrm{f}\left({x}\right)=\:\left(\mathrm{2}^{{x}} −\mathrm{1}\right)\left(\mathrm{2}^{{x}} −\mathrm{2}\right)^{\mathrm{2}} \\ $$

Question Number 36946 Answers: 0 Comments: 2

calculateϕ(λ)= ∫_0 ^π ((cos(t))/(1−2λ cost +λ^2 )) dt

$${calculate}\varphi\left(\lambda\right)=\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\:\:\frac{{cos}\left({t}\right)}{\mathrm{1}−\mathrm{2}\lambda\:{cost}\:+\lambda^{\mathrm{2}} }\:{dt} \\ $$

Question Number 36945 Answers: 0 Comments: 0

calulate ∫_0 ^(π/4) (dx/(√(tan(x)(1−tanx))))

$${calulate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{dx}}{\sqrt{{tan}\left({x}\right)\left(\mathrm{1}−{tanx}\right)}} \\ $$

Question Number 36944 Answers: 1 Comments: 1

find ϕ(a) = ∫_a ^(+∞) (dx/((1+x^2 )(√(x^2 −a^2 )))) with a>0

$${find}\:\varphi\left({a}\right)\:=\:\int_{{a}} ^{+\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−{a}^{\mathrm{2}} }}\:\:\:{with}\:{a}>\mathrm{0} \\ $$

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