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Question Number 33036 by rahul 19 last updated on 09/Apr/18

$$Ifrangeof$$$$f\left(x\right)=(x+1)(x+2)(x+3)(x+4)+5$$$$x\in [-6,6]is[a,b],a,b\in N,finda+b?$$

Answered by MJS last updated on 09/Apr/18

$$\mathrm{we}\mathrm{need}\mathrm{the}\mathrm{minimum}\mathrm{and}$$$$\mathrm{maximum}\mathrm{of}f\left(x\right)\mathrm{in}\mathrm{the}\mathrm{given}$$$$\mathrm{interval}$$$$f\left(x\right)=x^4+10x^3+35x^2+50x+29$$$$f^{\prime }\left(x\right)=4x^3+30x^2+70x+50=$$$$=2(2x+5)(x^2+5x+5)$$$$f^{\prime }\left(x\right)=0$$$$x_1=-\frac{5}{2}=-2.5$$$$x_2=-\frac{5}{2}-\frac{\sqrt{5}}{2}\approx -3.618$$$$x_3=-\frac{5}{2}+\frac{\sqrt{5}}{2}\approx -1.382$$$$f^{″}\left(x\right)=12x^2+60x+70$$$$f^{″}\left(x_1\right)<0\Rightarrow x_1\mathrm{is}\mathrm{local}\mathrm{maximum}$$$$f^{″}\left(x_2\right)>0\Rightarrow x_2\mathrm{is}\mathrm{local}\mathrm{minimum}$$$$f^{″}\left(x_3\right)>0\Rightarrow x_3\mathrm{is}\mathrm{local}\mathrm{minimum}$$$$f\left(x_1\right)=\frac{89}{16}=5.5625<f(-6)<f\left(6\right)$$$$f\left(x_2\right)=f\left(x_3\right)=4$$$$$$$$max\left(f\left(x\right)\right)=f\left(6\right)=5045;x\in [-6;6]$$$$min\left(f\left(x\right)\right)=f(-\frac{5}{2}\pm \frac{\sqrt{5}}{2})=4;x\in \mathbb{R}$$$$5045+4=5049$$

Commented by rahul 19 last updated on 09/Apr/18

$$iamnotabletounderstandlast2lines;$$$$whatif5.5625>f\left(6\right)?andifitlies$$$$inbetweenf(-6),f\left(6\right)?$$$$similarlywehavenotputf(-6)$$$$asitdoesnotgivemin.value?$$$$whyx\in [-6,6]andx\in Rothertime?$$

Commented by MJS last updated on 09/Apr/18

$$\mathrm{in}\mathrm{this}\mathrm{case},f\left(6\right)=5045\mathrm{is}\mathrm{the}$$$$\mathrm{maximum}\mathrm{for}x\in [-6;6]$$$$\mathrm{we}\mathrm{found}\mathrm{local}\mathrm{maximum}\mathrm{at}$$$$x=-\frac{5}{2}\mathrm{which}could\mathrm{be}\mathrm{the}$$$$\mathrm{maximum}\mathrm{in}[-6;6]\mathrm{but}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{not}$$$$\mathrm{because}f(-\frac{5}{2})=\frac{89}{16}<5045$$$$\mathrm{so}\mathrm{the}\mathrm{maximum}\mathrm{of}f\left(x\right)\mathrm{im}\mathrm{this}$$$$\mathrm{interval}\mathrm{is}5045$$$$\mathrm{the}\mathrm{minimum}\mathrm{is}4,{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{even}\mathrm{the}$$$$\mathrm{absolute}\mathrm{minimum}\mathrm{of}f\left(x\right)\mathrm{for}$$$$x\in \mathbb{R}\Rightarrow {\mathrm{it}}^{\prime }\mathrm{s}\mathrm{also}\mathrm{the}\mathrm{minimum}$$$$\mathrm{within}[-6;6]$$

Commented by rahul 19 last updated on 09/Apr/18

$$thankusomuchsir!$$

Commented by MJS last updated on 10/Apr/18

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{welcome}$$

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