Menu Close

Question-45482




Question Number 45482 by Meritguide1234 last updated on 13/Oct/18
Answered by ajfour last updated on 13/Oct/18
f(x)=((1−2x)/7)  ;    f(4)= −1   _________________________  let   f(x)=ax^2 +bx+c+1  ∫_0 ^(  x) f(x)dx= ((ax^3 )/3)+((bx^2 )/2)+(c+1)x  ⇒  a=(a/3)+(b/2)+(c+1)    ....(i)         b = ((8a)/3)+2b+2(c+1)       ...(ii)        c+1= 9a+((9b)/2)+3(c+1)+1   ..(iii)  solving eq. (i), (ii), (iii)   it is  found  that   a=0 , b = −(2/7)      and    c+1 = (1/7)   so    f(x)= ((1−2x)/7) ;  f(4)= −1 .
$${f}\left({x}\right)=\frac{\mathrm{1}−\mathrm{2}{x}}{\mathrm{7}}\:\:;\:\:\:\:{f}\left(\mathrm{4}\right)=\:−\mathrm{1}\: \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$${let}\:\:\:{f}\left({x}\right)={ax}^{\mathrm{2}} +{bx}+{c}+\mathrm{1} \\ $$$$\int_{\mathrm{0}} ^{\:\:{x}} {f}\left({x}\right){dx}=\:\frac{{ax}^{\mathrm{3}} }{\mathrm{3}}+\frac{{bx}^{\mathrm{2}} }{\mathrm{2}}+\left({c}+\mathrm{1}\right){x} \\ $$$$\Rightarrow\:\:{a}=\frac{{a}}{\mathrm{3}}+\frac{{b}}{\mathrm{2}}+\left({c}+\mathrm{1}\right)\:\:\:\:….\left({i}\right) \\ $$$$\:\:\:\:\:\:\:{b}\:=\:\frac{\mathrm{8}{a}}{\mathrm{3}}+\mathrm{2}{b}+\mathrm{2}\left({c}+\mathrm{1}\right)\:\:\:\:\:\:\:…\left({ii}\right) \\ $$$$\:\:\:\:\:\:{c}+\mathrm{1}=\:\mathrm{9}{a}+\frac{\mathrm{9}{b}}{\mathrm{2}}+\mathrm{3}\left({c}+\mathrm{1}\right)+\mathrm{1}\:\:\:..\left({iii}\right) \\ $$$${solving}\:{eq}.\:\left({i}\right),\:\left({ii}\right),\:\left({iii}\right)\:\:\:{it}\:{is} \\ $$$${found}\:\:{that}\:\:\:\boldsymbol{{a}}=\mathrm{0}\:,\:\boldsymbol{{b}}\:=\:−\frac{\mathrm{2}}{\mathrm{7}} \\ $$$$\:\:\:\:{and}\:\:\:\:\boldsymbol{{c}}+\mathrm{1}\:=\:\frac{\mathrm{1}}{\mathrm{7}}\: \\ $$$$\boldsymbol{{so}}\:\:\:\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\:\frac{\mathrm{1}−\mathrm{2}\boldsymbol{{x}}}{\mathrm{7}}\:;\:\:\boldsymbol{{f}}\left(\mathrm{4}\right)=\:−\mathrm{1}\:. \\ $$
Commented by Meritguide1234 last updated on 13/Oct/18
Commented by Meritguide1234 last updated on 13/Oct/18
very nice solution
$${very}\:{nice}\:{solution} \\ $$
Commented by rahul 19 last updated on 13/Oct/18
perfect solution. @Meritguide keep posting a problem a day in Calculus .( definite integration) I like your problems.

Leave a Reply

Your email address will not be published. Required fields are marked *