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Question-58816




Question Number 58816 by Tawa1 last updated on 30/Apr/19
Answered by Kunal12588 last updated on 30/Apr/19
from looking the graph we can say that  maxima of function is at  x=2  for A)  ((d(f(x)))/dx)=0     [for minima or maxima]  −(1/3)×2(x−2)(1)=0  ⇒x=2  option A can be correct  for B)  ((d(f(x)))/dx)=0  −(1/3)×2(x+2)(1)=0  ⇒x=−2  option B is not correct  for C)  ((d(f(x)))/dx)=0  (1/3)×2(x+2)(1)=0  ⇒x=−2  option C is not correct  for D)  ((d(f(x)))/dx)=0  6(x−2)(1)=0  ⇒x=2  ∴ only A or D options can be correct  now lets find the zeroes of A and D  A.   −(1/3)(x−2)^2 +5=0  ⇒(x−2)=±(√(−(5)(−3)))  ⇒x=2±(√(15))  D.    3(x−2)^2 +5=0    ⇒(x−2)=±(√(−(5/3)))  ⇒x=2±z  ⇒x∉ R  But zeroes of the equation in the graph are Real  ∴ Only A is correct  Ans. A. f(x)=−(1/3)(x−2)^2 +5
$${from}\:{looking}\:{the}\:{graph}\:{we}\:{can}\:{say}\:{that} \\ $$$${maxima}\:{of}\:{function}\:{is}\:{at}\:\:{x}=\mathrm{2} \\ $$$$\left.{for}\:{A}\right) \\ $$$$\frac{{d}\left({f}\left({x}\right)\right)}{{dx}}=\mathrm{0}\:\:\:\:\:\left[{for}\:{minima}\:{or}\:{maxima}\right] \\ $$$$−\frac{\mathrm{1}}{\mathrm{3}}×\mathrm{2}\left({x}−\mathrm{2}\right)\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\Rightarrow{x}=\mathrm{2} \\ $$$${option}\:{A}\:{can}\:{be}\:{correct} \\ $$$$\left.{for}\:{B}\right) \\ $$$$\frac{{d}\left({f}\left({x}\right)\right)}{{dx}}=\mathrm{0} \\ $$$$−\frac{\mathrm{1}}{\mathrm{3}}×\mathrm{2}\left({x}+\mathrm{2}\right)\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\Rightarrow{x}=−\mathrm{2} \\ $$$${option}\:{B}\:{is}\:{not}\:{correct} \\ $$$$\left.{for}\:{C}\right) \\ $$$$\frac{{d}\left({f}\left({x}\right)\right)}{{dx}}=\mathrm{0} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}×\mathrm{2}\left({x}+\mathrm{2}\right)\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\Rightarrow{x}=−\mathrm{2} \\ $$$${option}\:{C}\:{is}\:{not}\:{correct} \\ $$$$\left.{for}\:{D}\right) \\ $$$$\frac{{d}\left({f}\left({x}\right)\right)}{{dx}}=\mathrm{0} \\ $$$$\mathrm{6}\left({x}−\mathrm{2}\right)\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\Rightarrow{x}=\mathrm{2} \\ $$$$\therefore\:{only}\:{A}\:{or}\:{D}\:{options}\:{can}\:{be}\:{correct} \\ $$$${now}\:{lets}\:{find}\:{the}\:{zeroes}\:{of}\:{A}\:{and}\:{D} \\ $$$${A}.\:\:\:−\frac{\mathrm{1}}{\mathrm{3}}\left({x}−\mathrm{2}\right)^{\mathrm{2}} +\mathrm{5}=\mathrm{0} \\ $$$$\Rightarrow\left({x}−\mathrm{2}\right)=\pm\sqrt{−\left(\mathrm{5}\right)\left(−\mathrm{3}\right)} \\ $$$$\Rightarrow{x}=\mathrm{2}\pm\sqrt{\mathrm{15}} \\ $$$${D}.\:\:\:\:\mathrm{3}\left({x}−\mathrm{2}\right)^{\mathrm{2}} +\mathrm{5}=\mathrm{0}\:\: \\ $$$$\Rightarrow\left({x}−\mathrm{2}\right)=\pm\sqrt{−\frac{\mathrm{5}}{\mathrm{3}}} \\ $$$$\Rightarrow{x}=\mathrm{2}\pm{z} \\ $$$$\Rightarrow{x}\notin\:\mathbb{R} \\ $$$${But}\:{zeroes}\:{of}\:{the}\:{equation}\:{in}\:{the}\:{graph}\:{are}\:{Real} \\ $$$$\therefore\:{Only}\:{A}\:{is}\:{correct} \\ $$$${Ans}.\:{A}.\:{f}\left({x}\right)=−\frac{\mathrm{1}}{\mathrm{3}}\left({x}−\mathrm{2}\right)^{\mathrm{2}} +\mathrm{5} \\ $$
Commented by Kunal12588 last updated on 30/Apr/19
Commented by tanmay last updated on 30/Apr/19
bah excellent...very good..
$${bah}\:{excellent}…{very}\:{good}.. \\ $$
Commented by Tawa1 last updated on 30/Apr/19
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$
Commented by Kunal12588 last updated on 30/Apr/19
It was my pleasure. It is my first try of   such questions.
$${It}\:{was}\:{my}\:{pleasure}.\:{It}\:{is}\:{my}\:{first}\:{try}\:{of}\: \\ $$$${such}\:{questions}. \\ $$
Answered by MJS last updated on 30/Apr/19
it′s a hanging parabola ⇒  ⇒ f(x)=ax^2 +bx+c with a<0 ⇒  ⇒ options C, D don′t fit  A  x^2 −4x−11=0 ⇒ −(p/2)=2 ⇒ max at 2  B  x^2 +4x−11=0 ⇒ −(p/2)=−2 ⇒ max at −2  ⇒ option A
$$\mathrm{it}'\mathrm{s}\:\mathrm{a}\:\mathrm{hanging}\:\mathrm{parabola}\:\Rightarrow \\ $$$$\Rightarrow\:{f}\left({x}\right)={ax}^{\mathrm{2}} +{bx}+{c}\:\mathrm{with}\:{a}<\mathrm{0}\:\Rightarrow \\ $$$$\Rightarrow\:\mathrm{options}\:\mathrm{C},\:\mathrm{D}\:\mathrm{don}'\mathrm{t}\:\mathrm{fit} \\ $$$$\mathrm{A}\:\:{x}^{\mathrm{2}} −\mathrm{4}{x}−\mathrm{11}=\mathrm{0}\:\Rightarrow\:−\frac{{p}}{\mathrm{2}}=\mathrm{2}\:\Rightarrow\:\mathrm{max}\:\mathrm{at}\:\mathrm{2} \\ $$$${B}\:\:{x}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{11}=\mathrm{0}\:\Rightarrow\:−\frac{{p}}{\mathrm{2}}=−\mathrm{2}\:\Rightarrow\:\mathrm{max}\:\mathrm{at}\:−\mathrm{2} \\ $$$$\Rightarrow\:\mathrm{option}\:{A} \\ $$
Commented by Tawa1 last updated on 30/Apr/19
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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