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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 26, 2023
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What is the smallest distance?

 

Hello Guest!

 

\(y = 0.5\cdot (x^2 - 18)\\ y=0.5x^2-9\\ r^2(x)=x^2+y^2\\ r^2(x)=x^2+(0.5x^2-9)^2\\ \frac{dr^2(x)}{dx}=2x+2x(0.5x^2-9)\\ \frac{dr^2(x)}{dx}=2x+x^3-18x\)

\( \frac{dr^2(x)}{dx}=x^3-16x=x(x^2-16)=0\\ x\in \{-4,0,{\color{blue}4}\}\\ y=0.5\cdot 16-9\\ \color{blue}y=-1\)

\(l=\sqrt{x^2+y^2}=\sqrt{16+1}\\ \color{blue}l=4.123\)

 

The smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 18) is 4.123 .

laugh  !

 Apr 26, 2023

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