What is the smallest distnace between the origin and a point on the graph of \(y = \frac{1}{\sqrt{2}} (x^2 - 18)\)?
+15001 What is the smallest distance between the origin and a point on the graph?
Hello Guest!
\(\color{blue}f(x)= y=\dfrac{1}{\sqrt{2}} (x^2 - 18)\\ y= \dfrac{x^2}{\sqrt{2}}-9\cdot \sqrt{2}\\ \color{blue}\dfrac{dy}{dx}=y\ '=\sqrt{2}\cdot x\)
\(g(x)=-\frac{1}{y\ '}(x-0)+0\\ g(x)=-\frac{1}{\sqrt{2}\cdot x}(x-0)+0\ |\ point\ direction\ equation\\ \color{blue}g(x)= -\frac{1}{\sqrt{2}}\)
\(\color{blue}g(x)=f(x)\\-\dfrac{1}{\sqrt{2}}=\dfrac{1}{\sqrt{2}} (x^2 - 18)\\ x^2-18+1=0\\ x= \sqrt{17}\)
\(x=4.1231\)
\( y=\dfrac{1}{\sqrt{2}} (x^2 - 18)\\ y=\dfrac{1}{\sqrt{2}} (17 - 18)=-\dfrac{1}{\sqrt{2}}\\\)
\(y=0.7071\)
\(d=\sqrt{x^2+y^2}=\sqrt{17+\frac{1}{2}}\)
\(d=4.1833\)
The smallest distance between the origin and a point on the graph is 4.1833 .
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