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Question Number 86894 by M±th+et£s last updated on 01/Apr/20

$$letf\left(x\right)=\frac{1-x^2}{\lceil x-1\rceil }$$$$findthedomainandf{}^{\prime }(-\frac{1}{2})ifexist.$$$$$$$$\lceil ...\rceil ceilingfunction$$

Commented by mathmax by abdo last updated on 01/Apr/20

$$if[..]meanintegrpartwehavef\left(x\right)=\frac{1-x^2}{\left[x\right]-1}$$$$let\left[x\right]=p\Rightarrow p⩽x<p+1\Rightarrow f\left(x\right)=\frac{1-x^2}{p-1}andforp\ne 1weget$$$$f^{{}^{\prime }}\left(x\right)=\frac{1}{p-1}(-2x)=\frac{-2x}{p-1}$$$$x=-\frac{1}{2}\Rightarrow [-\frac{1}{2}]=-1\Rightarrow f^{{}^{\prime }}(-\frac{1}{2})=\frac{(-2)\times (-\frac{1}{2})}{-2}=-\frac{1}{2}$$$$domainoff$$$$\left[x\right]-1=0\iff \left[x\right]=1\Rightarrow 1⩽x<2\Rightarrow D_f=R-[1,2[$$

Commented by M±th+et£s last updated on 01/Apr/20

$$\lceil ...\rceil smallestintegerfunction$$

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