Let-f-R-R-be-a-function-such-that-f-x-x-3-x-2-f-1-xf-2-f-3-for-x-R-1-What-is-f-1-equal-to-2-What-is-f-1-equal-to-3-What-is-f-10-equal-to-For-this-question-consider-the-following-

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Question Number 10118 by Gaurav3651 last updated on 25/Jan/17

$$Letf:\mathbb{R}\to \mathbb{R}beafunctionsuchthat$$$$f\left(x\right)=x^3+x^2{f}^{\prime }\left(1\right)+{xf}^{″}\left(2\right)+f^{‴}\left(3\right)$$$$forx\in \mathbb{R}.$$$$1)Whatisf\left(1\right)equalto?$$$$2)Whatis{f}^{\prime }\left(1\right)equalto?$$$$3)Whatis{f}^{‴}\left(10\right)equalto?$$$$Forthisquestionconsiderthefollowing:$$$$1)f\left(2\right)=f\left(1\right)-f\left(0\right)$$$$2)f^{″}\left(2\right)-2f^{\prime }\left(1\right)=12$$$$whichis/arecorrect?$$

Commented by nume1114 last updated on 25/Jan/17

$$Answer$$$$1)f\left(1\right)=4$$$$2)f^{\prime }\left(1\right)=-5$$$$3)f^{‴}\left(10\right)=6$$$$$$$$1)f\left(2\right)=f\left(1\right)-f\left(0\right)$$$$2)f^{″}\left(2\right)-2f^{\prime }\left(1\right)=12$$$$\mathrm{Both}\mathrm{of}\mathrm{these}\mathrm{are}\mathrm{correct}.$$$$$$

Answered by nume1114 last updated on 25/Jan/17

$$f\left(x\right)=x^3+x^2{f}^{\prime }\left(1\right)+{xf}^{″}\left(2\right)+f^{‴}\left(3\right)\dots \left(A\right)$$$$f^{\prime }\left(x\right)=3x^2+2{xf}^{\prime }\left(1\right)+f^{″}\left(2\right)\dots \left(B\right)$$$$f^{″}\left(x\right)=6x+2f^{\prime }\left(1\right)\dots \left(C\right)$$$$f^{‴}\left(x\right)=6\dots \left(D\right)$$$$\left(D\right)\Rightarrow f^{‴}\left(3\right)=6\dots \left(E\right)$$$$\Rightarrow f^{‴}\left(10\right)=6$$$$\left(B\right)\Rightarrow f^{\prime }\left(1\right)=3+2f^{\prime }\left(1\right)+f^{″}\left(2\right)$$$$\Rightarrow f^{\prime }\left(1\right)+f^{″}\left(2\right)=-3\dots \left(F\right)$$$$\left(C\right)\Rightarrow f^{″}\left(2\right)=12+2f^{\prime }\left(1\right)\dots \left(G\right)$$$$\Rightarrow f^{″}\left(2\right)-2f^{\prime }\left(1\right)\stackrel{!}{=}12$$$$\left(F\right),\left(G\right)\Rightarrow f^{\prime }\left(1\right)+[12+2f^{\prime }\left(1\right)]=-3$$$$\Rightarrow f^{\prime }\left(1\right)=-5\dots \left(H\right)$$$$\Rightarrow f^{″}\left(2\right)=2\dots \left(I\right)$$$$\left(A\right),\left(E\right),\left(H\right),\left(I\right)$$$$\Rightarrow f\left(x\right)=x^3-5x^2+2x+6$$$$\Rightarrow f\left(1\right)=4$$$$\Rightarrow f\left(2\right)=-2$$$$\Rightarrow f\left(0\right)=6$$$$\Rightarrow f\left(2\right)\stackrel{!}{=}f\left(1\right)-f\left(0\right)$$

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