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Question Number 45482 by Meritguide1234 last updated on 13/Oct/18

Answered by ajfour last updated on 13/Oct/18

$$f\left(x\right)=\frac{1-2x}{7};f\left(4\right)=-1$$$$\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}$$$$letf\left(x\right)={ax}^2+bx+c+1$$$${\int }_0^{x}f\left(x\right)dx=\frac{{ax}^3}{3}+\frac{{bx}^2}{2}+(c+1)x$$$$\Rightarrow a=\frac{a}{3}+\frac{b}{2}+(c+1)....\left(i\right)$$$$b=\frac{8a}{3}+2b+2(c+1)...\left(ii\right)$$$$c+1=9a+\frac{9b}{2}+3(c+1)+1..\left(iii\right)$$$$solvingeq.\left(i\right),\left(ii\right),\left(iii\right)itis$$$$foundthat\mathit{a}=0,\mathit{b}=-\frac{2}{7}$$$$and\mathit{c}+1=\frac{1}{7}$$$$\mathit{s}\mathit{o}\mathit{f}\left(\mathit{x}\right)=\frac{1-2\mathit{x}}{7};\mathit{f}\left(4\right)=-1.$$

Commented by Meritguide1234 last updated on 13/Oct/18

Commented by Meritguide1234 last updated on 13/Oct/18

$$verynicesolution$$

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.

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