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What is the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 18)?

 Apr 30, 2023
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 the smallest distance

 

\(y=0.5(x^2-18)\\ l^2=x^2+y^2\\ l^2=x^2+0.5^2\cdot (x^2-18)^2\\ \dfrac{d\ l^2}{dx}=2x+0.25\cdot 2\cdot (x^2-18)\cdot 2x=0\\ 2x+x(x^2-18)=0\\ 2+x^2-18=0\\ x^2=16\\ x\in \{-4,4\}\)

\(l=\sqrt{x^2+y^2}=\sqrt{4^2+(-1)^2}\\ \color{blue}l=\sqrt{17}=4.123\)

Tthe smallest distance is 4.123.

laugh  !

 Apr 30, 2023
edited by asinus  Apr 30, 2023
edited by asinus  Apr 30, 2023

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