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Question Number 46552 by peter frank last updated on 28/Oct/18

Commented by MJS last updated on 28/Oct/18

$$b)\mathrm{is}\mathrm{not}\mathrm{clear}.\mathrm{what}\mathrm{does}\mathrm{this}\mathrm{mean}‘‘\mathrm{if}(\mathrm{it}?)$$$$\mathrm{is}\mathrm{parallel}\mathrm{to}y=2x+3\mathrm{at}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^{″}?\mathrm{if}\left(\begin{array}{c}0\\ 1\end{array}\right)\mathrm{is}\mathrm{a}\mathrm{point}$$$$\mathrm{on}\mathrm{the}\mathrm{curve}\left(\begin{array}{c}0\\ 4\end{array}\right)\mathrm{cannot}\mathrm{be}\mathrm{or}\mathrm{vice}\mathrm{versa}.$$$$\mathrm{please}\mathrm{check}\mathrm{this}$$

Commented by peter frank last updated on 28/Oct/18

$$\mathrm{yes}\mathrm{sir}!\mathrm{definitely}\mathrm{problem}\mathrm{is}\mathrm{not}\mathrm{clear}$$

Answered by MJS last updated on 28/Oct/18

$$f\left(x\right)=x^3-3x^2+x+4$$$$\mathrm{in}\mathrm{turning}\mathrm{points}{f}^{″}\left(x\right)=0$$$$\mathrm{polynomes}\mathrm{of}{3}^{\mathrm{rd}}\mathrm{degree}\mathrm{have}1\mathrm{turning}\mathrm{point}$$$$f^{″}\left(x\right)=6x-6$$$$\Rightarrow \mathrm{turning}\mathrm{point}T=\left(\begin{array}{c}1\\ 3\end{array}\right)$$$$\mathrm{checking}\mathrm{the}\mathrm{tangent}\mathrm{in}\mathrm{this}\mathrm{point}$$$$f^{\prime }\left(x\right)=3x^2-6x+1$$$$f^{\prime }\left(1\right)=-2\Rightarrow \mathrm{tangent}\mathrm{decreasing}\Rightarrow$$$$\Rightarrow \mathrm{curvature}\mathrm{negative}\left(\mathrm{clockwise}\right)\mathrm{for}x<1$$$$\mathrm{curvature}\mathrm{positive}\left(\mathrm{counterclockwise}\right)\mathrm{for}x>1$$

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