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Question Number 169631 by MaxiMaths last updated on 05/May/22

Commented by MaxiMaths last updated on 05/May/22

$$\mathrm{any}\mathrm{way}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{function}f???$$

Commented by mr W last updated on 05/May/22

$$youmeant-4atx=-1?$$

Commented by mr W last updated on 05/May/22

$$infinitepossibilities.$$$$examplef\left(x\right)=-\frac{14}{9}(\frac{x^3}{3}-\frac{x^2}{2}-2x)-\frac{59}{27}$$

Commented by mr W last updated on 05/May/22

Answered by mr W last updated on 05/May/22

$$f^{\prime }\left(x\right)=g\left(x\right)(x+1)(x-2)$$$$f\left(x\right)=\int g\left(x\right)(x+1)(x-2)+C$$$$f(-1)=-4,f\left(2\right)=3$$$$\Rightarrow thereareinfinitepossibiltiesfor$$$$g\left(x\right)andC.$$$$$$$$themostsimplecaseis$$$$g\left(x\right)=a=constant.$$$$$$$$f^{\prime }\left(x\right)=a(x+1)(x-2)=a(x^2-x-2)$$$$f\left(x\right)=a(\frac{x^3}{3}-\frac{x^2}{2}-2x)+b$$$$f(-1)=a(-\frac{1}{3}-\frac{1}{2}+2)+b=-4$$$$\Rightarrow \frac{7}{6}a+b=-4$$$$f\left(2\right)=a(\frac{8}{3}-\frac{4}{2}-4)+b=3$$$$\Rightarrow -\frac{10}{3}a+b=3$$$$\Rightarrow a=-\frac{14}{9}$$$$\Rightarrow b=-\frac{59}{27}$$$$\Rightarrow f\left(x\right)=-\frac{14}{9}(\frac{x^3}{3}-\frac{x^2}{2}-2x)-\frac{59}{27}$$

Commented by MaxiMaths last updated on 06/May/22

$$\mathrm{waoh}!\mathrm{good}!$$

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