#1**0 **

If we graph it on desmos, just to get a visual,

We can see that the x intercepts are the closest point. Plugging in y = 0,

\(0=\frac{1}{\sqrt{2}}\left(x^{2}-3\right) \\ \frac{x^2}{\sqrt{2}} - \frac{3}{\sqrt{2}} = 0 \\ x^2 - 3 = 0 \\ x^2 = 3 \\ x = \pm\sqrt{3}\)

So the smallest distance is $\sqrt{3}$

Awesomeguy Jul 8, 2021

#2**0 **

you can't always jump to conclusions like that just by looking at the graph. :)

anyway

the distance from the graph to the origin can be represented like this (by the Pythagorean theorem):

\(\sqrt{x^2+(\frac{1}{\sqrt{2}}(x^2-3))^2}\\=\sqrt{x^2+\frac{(x^2-3)^2}{2}}\\= \sqrt{x^2+\frac{x^4-6x^2+9}{2}}\\=\sqrt{\frac{x^4-4x^2+9}{2}}\\ =\sqrt{\frac{x^4-4x^2+4+9-4}{2}}\\=\sqrt{\frac{(x^2-2)^2+5}{2}}\)

Since \((x^2-2)^2\) will always be nonnegative for real numbers, the best you can do to minimize it is to set it equal to 0, and you will obtain your final answer:

\(\sqrt{\frac{0+5}{2}}=\boxed{\sqrt{\frac{5}{2}}}\)

for completeness, if you want to know the x value for which this minimum distance occurs, you can just solve an equation:

\((x^2-2)^2=0\\x^2-2=0\\x=\pm\sqrt{2}\)

textot
Jul 8, 2021